I am addressing only the first part of your question (i.e., nothing with the structure of vector space; only topology and limits of sequences).
I will quote here part of Problems 1.7.18-1.7.20 from Engelking's General Topology. (It would be better if you could get the book. I believe it used to be here, but the links don't work now. Perhaps you'll find it in the Internet.)
L*-space is a pair $(X, \lambda)$, where X is
a set and $\lambda$ a function (called the limit operator) assigning to some sequences of points of X
an element of X (called the limit of the sequence) in such a way that the following conditions are satisfied:
(L1) If $x_i=x$ for $i = 1,2,\dots$, then $\lambda x_i = x$.
(L2) If $\lambda x_i = x$, then $\lambda x' = x$ for every subsequence $x'$ of $x$.
(L3) If a sequence $\{x_i\}$ does not converge to $x$, then it contains a subsequence $\{x_{k_i}\}$ such
that no subsequence of $\{x_{k_i}\}$ converges to $x$.
These properties are sufficient to define a closure operator on $X$ (not necessary idempotent).
If $(X,\lambda)$ fulfills and additional condition
(L4) If $\lambda x_i = x$ and $\lambda x^i_j = x_i$ for $i = 1,2,\dots$, then there exist sequences of positive integers
$i_1, i_2,\dots$ and $j_1, j_2, \dots$ such that $\lambda x_{j_k}^{i_k} = x$.
L*-space $X$ satisfying (L4) is called an S*-space. The closure operator given by S*-space is idempotent.
Using this closure operator we get a topology, such that the convergence of the sequences is given by $\lambda$. A topology can be obtained from a L*-space (S*-space) if and only if the original space is sequential (Frechet-Urysohn).
References given in Engelking's book are Frechet [1906] and [1918], Urysohn [1926a], Kisynski [i960].
Frechet [1906] Sur quelques points du calcul fonctionnel, Rend, del Circ. Mat. di Palermo 22 (1906), 1-74.
Frechet [1918] Sur la notion de voisinage dans les ensembles abstraits, Bull. Sci. Math. 42 (1918), 138-156.
Kisynski [1960] Convergence du type L, Coll. Math. 7 (1960), 205-211.
Urysohn [1926a] Sur les classes (L) de M. Frechet, Enseign. Math. 25 (1926), 77-83.
NOTE: Some axioms for convergence of sequences are studied in the paper:
Mikusinski, P., Axiomatic theory of convergence (Polish), Uniw. Slaski w Katowicach Prace Nauk.-Prace Mat. No. 12 (1982), 13-21. I do not have the original paper, only a paper which cites this one; it seems that the axioms are equivalent to (L1)-(L3) and the uniqueness of limit. (But I do not know, whether some further axioms are studied in this paper.)
EDIT: In Engelking's book (and frequently in general topology) the term Frechet space is used in this sense, not this one. I've edited Frechet to Frechet-Urysohn above, to avoid the confusion.
Best Answer
I'm answering this in case anyone else arrives at this question. Write $(X,\tau_X)$ for the topology of $X$, and we define a topology $\tau_{LU}$ of locally uniform convergence on $C(X,\mathbb R)$ as follows.
Let $\mathcal B\subset \tau_X$ be a neighborhood basis of $(X,\tau_X)$ which is closed under finite unions. For any $f\in C(X,\mathbb R)$, any $U\in\mathcal B$, and any $\varepsilon>0$, we define the open $$ \mathscr U_{f,U,\varepsilon} := \{g\in C(X):\sup_U|f-g|<\varepsilon\}. $$ We now show that the family $\{\mathscr U_{f,U,\varepsilon}\}_{f\in C(X),U\in\mathcal B,\varepsilon>0}$ forms a neighborhood basis for $C(X)$. Suppose $g\in\mathscr U_{f_1,U_1,\varepsilon_1}\cap \mathscr U_{f_2,U_2,\varepsilon_2}$. Setting $\varepsilon_i' = \varepsilon_i - \sup_{U_i}|f_i-g|>0$ and letting $\varepsilon:=\min\{\varepsilon_1',\varepsilon_2'\}$, we get that $$ g \in \mathscr U_{g,U_1\cup U_2,\varepsilon}\subset \mathscr U_{f_1,U_1,\varepsilon_1}\cap \mathscr U_{f_2,U_2,\varepsilon_2}. $$ We call the resulting topology $\tau_{\mathcal B}$.
We can now define the topology $$ \tau_{LU} = \bigcap_{\mathcal B}\tau_{\mathcal B} $$ where we index over neighborhood bases $\mathcal B\subset\tau_X$ that are closed under finite unions. This gives the desired topology.