First, there's no need to focus on online copies, as asked for in the question. We used to have things called libraries which contain journal articles in them. :) Try looking there.
More seriously, I think your task is to a large extent hopeless. Most of those works were never translated into English. But there are numerous English language sources which describe some aspect of how class field theory was originally developed and you should start there.
Here are some:
G. Frei, Heinrich Weber and the Emergence of Class Field Theory, in ``The History of Modern Mathematics, vol. 1: Ideas and their Reception,'' (J. McCleary and D. E. Rowe, ed.) Academic Press, Boston, 1989, 424--450.
H. Hasse, ``Class Field Theory,'' Lecture Notes # 11, Dept. Math. Univ. Laval, Quebec, 1973. [This is basically adapted from his paper in Cassels and Frohlich, but has some nuggets that were not in C&F.]
K. Iwasawa, On papers of Takagi in Number Theory, in ``Teiji Takagi Collected Papers,'' 2nd ed., Springer-Verlag, Tokyo, 1990, 342--351.
S. Iyanaga, ``The Theory of Numbers,'' North-Holland, Amsterdam, 1975. [The end of the book has a nice exposition of how alg. number theory developed up to class field theory.]
S. Iyanaga, On the life and works of Teiji Takagi, in ``Teiji Takagi Collected Papers,'' 2nd ed., Springer-Verlag, Tokyo, 1990, 354--371.
S. Iyanaga, Travaux de Claude Chevalley sur la th\'eorie du corps de classes:
Introduction, Japan. J. Math. 1 (2006), 25--85. [Are you OK with French?]
M. Katsuya, The Establishment of the Takagi--Artin Class Field Theory, in
``The Intersection of History and Mathematics,'' (C. Sasaki, M. Sugiura, J. W. Dauben ed.),
Birkhauser, Boston, 1995, 109--128.
T. Masahito, Three Aspects of the Theory of Complex Multiplication,
``The Intersection of History and Mathematics,'' (C. Sasaki, M. Sugiura, J. W. Dauben ed.),
Birkhauser, Boston, 1995, 91--108.
K. Miyake, Teiji Takagi, Founder of the Japanese School of Modern Mathematics,
Japan. J. Math. 2 (2007), 151--164.
P. Roquette, Class Field Theory in Characteristic $p$, its Origin and Development,
in ``Class Field Theory -- its Centenary and Prospect,'' Math. Soc. Japan, Tokyo, 2001,
549--631.
H. Weyl, David Hilbert and His Mathematical Work, Bull. Amer. Math. Soc. {\bf 50} (1944), 612--654. [Hilbert's obituary]
I did write up something myself about a year or so ago on the history of class field theory just to put in one place what I was able to cobble together from these kinds of sources. See
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/cfthistory.pdf
which contains the above references as the bulk of the bibliography (I did not just type all those articles references above by hand!) The main thing which had baffled me at first was how they originally defined the local norm residue symbol at ramified primes. I give some examples of how this was determined in the original language of central simple algebras.
EDIT. Here is the part of the answer that has been rewritten:
We give below a short proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back at least to Emil Artin (precise references are given below).
The derivation of the FTGT from (a) and (b) takes about four lines, but I haven't been able to find these four lines in the literature, and all the proofs of the FTGT I have seen so far are much more complicated. So, if you find either a mistake in these four lines, or a trace of them the literature, please let me know.
The argument is essentially taken from Chapter II (link) of Emil Artin's Notre Dame Lectures [A]. More precisely, statement (a) below is implicitly contained in the proof Theorem 10 page 31 of [A], in which the uniqueness up to isomorphism of the splitting field of a polynomial is verified. Artin's proof shows in fact that, when the roots of the polynomial are distinct, the number of automorphisms of the splitting extension coincides with the degree of the extension. Statement (b) below is proved as Theorem 14 page 42 of [A]. The proof given here (using Artin's argument) was written with Keith Conrad's help.
Theorem. Let $E/F$ be an extension of fields, let $a_1,\dots,a_n$ be distinct generators of $E/F$ such that the product of the $X-a_i$ is in $F[X]$. Then
the group $G$ of automorphisms of $E/F$ is finite,
there is a bijective correspondence between the sub-extensions $S/F$ of $E/F$ and the subgroups $H$ of $G$, and we have
$$
S\leftrightarrow H\iff H=\text{Aut}(E/S)\iff S=E^H
$$
$$
\implies[E:S]=|H|,
$$
where $E^H$ is the fixed subfield of $H$, where $[E:S]$ is the degree (that is the dimension) of $E$ over $S$, and where $|H|$ is the order of $H$.
PROOF
We claim:
(a) If $S/F$ is a sub-extension of $E/F$, then $[E:S]=|\text{Aut}(E/S)|$.
(b) If $H$ is a subgroup of $G$, then $|H|=[E:E^H]$.
Proof that (a) and (b) imply the theorem. Let $S/F$ be a sub-extension of $E/F$ and put $H:=\text{Aut}(E/S)$. Then we have trivially $S\subset E^H$, and (a) and (b) imply
$$
[E:S]=[E:E^H].
$$
Conversely let $H$ be a subgroup of $G$ and set $\overline H:=\text{Aut}(E/E^H)$. Then we have trivially $H\subset\overline H$, and (a) and (b) imply $|H|=|\overline H|$.
Proof of (a). Let $1\le i\le n$. Put $K:=S(a_1,\dots,a_{i-1})$ and $L:=K(a_i)$. It suffices to check that any $F$-embedding $\phi$ of $K$ in $E$ has exactly $[L:K]$ extensions to an $F$-embedding $\Phi$ of $L$ in $E$; or, equivalently, that the polynomial $p\in\phi(K)[X]$ which is the image under $\phi$ of the minimal polynomial of $a_i$ over $K$ has $[L:K]$ distinct roots in $E$. But this is clear since $p$ divides the product of the $X-a_j$.
Proof of (b). In view of (a) it is enough to check $|H|\ge[E:E^H]$. Let $k$ be an integer larger than $|H|$, and pick a
$$
b=(b_1,\dots,b_k)\in E^k.
$$
We must show that the $b_i$ are linearly dependent over $E^H$, or equivalently that $b^\perp\cap(E^H)^k$ is nonzero, where $\bullet^\perp$ denotes the vectors orthogonal to $\bullet$ in $E^k$ with respect to the dot product on $E^k$. Any element of $b^\perp\cap (E^H)^k$ is necessarily orthogonal to $hb$ for any $h\in H$, so
$$
b^\perp\cap(E^H)^k=(Hb)^\perp\cap(E^H)^k,
$$
where $Hb$ is the $H$-orbit of $b$. We will show $(Hb)^\perp\cap(E^H)^k$ is nonzero. Since the span of $Hb$ in $E^k$ has $E$-dimension at most $|H| < k$, $(Hb)^\perp$ is nonzero. Choose a nonzero vector $x$ in $(Hb)^\perp$ such that $x_i=0$ for the largest number of $i$ as possible among all nonzero vectors in $(Hb)^\perp$. Some coordinate $x_j$ is nonzero in $E$, so by scaling we can assume $x_j=1$ for some $j$. Since the subspace $(Hb)^\perp$ in $E^k$ is stable under the action of $H$, for any $h$ in $H$ we have $hx\in(Hb)^\perp$, so $hx-x\in(Hb)^\perp$. Since $x_j=1$, the $j$-th coordinate of $hx-x$ is $0$, so $hx-x=0$ by the choice of $x$. Since this holds for all $h$ in $H$, $x$ is in $(E^H)^k$.
[A] Emil Artin, Galois Theory, Lectures Delivered at the University of Notre Dame, Chapter II, available here.
PDF version: http://www.iecl.univ-lorraine.fr/~Pierre-Yves.Gaillard/DIVERS/Selected_Texts/st.pdf
Here is the part of the answer that has not been rewritten:
Although I'm very interested in the history of Galois Theory, I know almost nothing about it. Here are a few things I believe. Thank you for correcting me if I'm wrong. My main source is
http://www-history.mcs.st-and.ac.uk/history/Projects/Brunk/Chapters/Ch3.html
Artin was the first mathematician to formulate Galois Theory in terms of a lattice anti-isomorphism.
The first publication of this formulation was van der Waerden's "Moderne Algebra", in 1930.
The first publications of this formulation by Artin himself were "Foundations of Galois Theory" (1938) and "Galois Theory" (1942).
Artin himself doesn't seem to have ever explicitly claimed this discovery.
Assuming all this is true, my (probably naive) question is:
Why does somebody who makes such a revolutionary discovery wait so many years before publishing it?
I also hope this is not completely unrelated to the question.
Best Answer
Takagi's goal is the following:
Step 1 is analogous to the construction of the fields of $p^n$-th roots of unity (in particular, proving the irreducibility of the corresponding cyclotomic polynomials and determining their dirscriminants), whereas Step 2 is the analogue of the theorem of Kronecker-Weber.
Step 1
Takagi (thesis, Sect. 6) constructs the following cyclic extensions of ${\mathbb Q}(i)$:
Takagi also proves the decomposition law in these extensions.
Step 2
In Sect. 9, Takagi begins proving the analogue of the theorem of Kronecker-Weber. His approach is the one used by Hilbert in his proof of the theorem of Kronecker-Weber: Given an abelian extension $L/{\mathbb Q}(i)$, we write $L$ as a compositum of cyclic extensions of prime power degree. If the odd Gaussian prime $\mu$ is ramified in $L$, form the compositum of $L$ and the cyclic extension $K/{\mathbb Q}(i)$ unramified outside $\mu$ constructed in Step 1 and show that the compositum $KL$ contains a cyclic extension $M/{\mathbb Q}(i)$ unramified at $\mu$ such that $L$ is contained in $KM$; the case of wild ramification requires a careful investiagtion of the ramification subgroups.
By induction, this process reduces the proof to cyclic extensions that are unramified everywhere. There are various ways of proving that such extensions do not exist:
We can use Takagi's Lemma, according to which every cyclic extension of ${\mathbb Q}(i)$ that is normal over ${\mathbb Q}$ is a cyclotomic field. This is how Takagi proves this claim.
By Hilbert's Theorem 94, unramified cyclic extensions of prime degree $\ell$ of a number field $K$ exist only if $K$ has class number divisible by $\ell$.
Minkowski's bounds show that ${\mathbb Q}(i)$ does not admit any nontrivial unramified extension.
P.S. Takagi's thesis is contained in his collected papers. I also would like to refer you to a beautiful article by Cox and Hyde on The Galois theory of the lemniscate.