Since $\epsilon_0$ is still an $\omega$-sequence, it seems to me that an embedding is relatively straightforward; take it as the sequence $\left\{\omega, \omega^\omega, \omega^{\omega^\omega}, \ldots\right\}$ and embed each of those in a unit interval. Each has an easy explicit embedding, and for any '$\omega$-polynomial' that you give me I can give you the rational in my embedding that corresponds to it. I suspect that things break down, as you say, not too many steps up in the hierarchy - but that starts to get to questions of what an 'explicit specification' is.
EDIT: And to answer the question about embedding $\omega^\omega$, the same can easily be done; to make it more straightforward and highlight my mapping above, I'll pack it into the unit interval $\left[1,2\right)$. Map $\omega$ onto $\left[1,1+\frac{1}{2}\right)$, $\omega^2$ onto $\left[1+\frac{1}{2}, 1+\frac{3}{4}\right)$, etc; then our effective procedure for 'decoding' a polynomial $\sum_{i=0}^na_i\omega^i$ to a rational produces the rational $1+(1-2^{-n})+2^{-(n+1)}\left(1-2^{-(a_n-1)}\right)+2^{-(n+1)}2^{-a_n}\left(1-2^{-(a_{n-1}-1)}\right)+\cdots$ — we 'chop off' the largest term so that we're working in the interval $\left[1+(1-2^{-n}), 1+(1-2^{-(n+1)})\right)$ of length $2^{-(n+1)}$, use $a_n$ to find the point in that interval representing $a_n\omega^n$ (and implicitly, the next 'mapped-down' interval), and repeat the procedure. As long as there's an explicit means of specifying an $\omega$-sequence for the ordinal, this general mechanism will work.
I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $\Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.
Regarding your definitions, the function $\phi_{\gamma}(\alpha)$ should also be greater than every ordinal $\phi_{\eta}^{\circ n}(\phi_{\gamma}(\beta)+1)$ for $\eta < \gamma$, $n \in \mathbb{N}$ and $\beta<\alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.
Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $\phi_{\gamma}$ can be extended to $\mathbf{No}$ in a natural way.
For $x=\{L\ | \ R\} \in \mathbf{No}$, you must know about $\omega^x=\phi_0(x)=\{0,\mathbb{N}\ \phi_0(L)\ | \ 2^{-\mathbb{N}} \ \phi_0(R)\}$.
Then the class of numbers $e$ such that $\omega^e=e$ is parametrized by $\varepsilon_x=\phi_1(x):=\{\phi_0^{\circ \mathbb{N}}(0),\phi_0^{\circ \mathbb{N}}(\phi_1(L)+1)\ | \ \phi_0^{\circ \mathbb{N}}(\phi_1(R)-1)\}$,
and one can keep going on. At every stage $0<\gamma$, the function $\phi_{\gamma}$ parametrizes the class of numbers $e$ with $\forall \eta < \gamma,\phi_{\eta}(e)=e$.
As for sources on $\mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.
edit: to be more explicit, Conway's so-called $\omega$-map is defined inductively as $x \longmapsto \omega^x:=\{0,n \ \omega^{x_L}:n \in \mathbb{N} \wedge x' \in x_L \ | \ 2^{-n} \ \omega^{x''}:n \in \mathbb{N} \wedge x'' \in x_R\}$ where $x=\{x_L \ | \ x_R\}$. This is done so as to yield $r \omega^x < s \omega^y$ whenever $x<y$ and $r,s$ are strictly positive real numbers.
For $\phi_1$, this is $\phi_1(x):=\{\phi_0^{\circ n}(0),\phi_0^{\circ n}(\phi_1(x')+1): x' \in x_L \wedge n \in \mathbb{N} \ | \ \phi_0^{\circ n}(\phi_1(x'')-1): x' \in x_R \wedge n \in \mathbb{N}\}$, where $f^{\circ n}$ denotes the $n$-fold composition of a function $f$ with itself.
You can find both of those in Conway's On Numbers and Games, Chapter 3 and in Gonshor's An Introduction to the Theory of Surreal Numbers, Chapters 5 and 9.
This is also discussed in some detail in Sections 5 and 6 of the pre-print Surreal Substructures (the formula for fixed points parametrizations is Remark 6.23).
Best Answer
The answer (unsurprisingly) depends on what you mean by "systematic way." Although $\omega_1^{CK}$ has no computable description, and so no "computable notation system" will reach up all the way to $\omega_1^{CK}$ (this can be made precise and proved), there are other ways to describe ordinals. For example, any $\alpha<\omega_1^{CK}$ has a natural number $n_\alpha$ assigned to it, namely the least Kleene notation corresponding to it. This is a perfectly well-defined description of $\alpha$; does that count?
In fact, the map $n:\omega_1^{CK}\rightarrow\omega:\alpha\mapsto n_\alpha$ turns out to be "as simple as possible" in a precise sense, and via this simplicity plays an important role in metarecursion theory (= $\omega_1^{CK}$-recursion theory). So it's not flippant to point out that it is "not too complicatedly definable" - that's actually something that matters!
On the other hand, $\omega_1^{CK}$ isn't the real barrier: the problem is that given any reasonable descriptive framework, there will be countable ordinals not describable in that way, simply because there are uncountably many countable ordinals and only countably many descriptions. "The least ordinal not described by system $S$" is going to be a perfectly reasonable definition of an ordinal, whenever $S$ is a perfectly reasonable descriptive framework, but clearly can't be part of $S$ itself.
On the other other hand, the word "reasonable" is doing some serious work there. There is a precise sense in which it is possible that every countable ordinal (indeed, everything whatsoever) is first-order definable in the universe. The reason this doesn't immediately lead to an explosion is that "(unrestricted) definability (within the whole universe) isn't definable", and an analysis of this bizarre situation can be found here.