According to König's theorem, the number of vertices in the minimum vertex cover is equal to the number of edges in the maximum matching of any bipartite graph. Thus, we can find the size of the minimum vertex cover by finding the maximum matching of the graph (actually finding the vertices is possible but harder).

This is a standard problem that can be solved using maximum flow algorithms. If you don't know about maximum flow, you can read about it in the Competitive Programmer's Handbook — Chapter 20 (20.1 and 20.3). For implementation of the Ford-Fulkerson and Edmonds-Karp-algorithms, you can see this page from CP-Algorithms.

You can also see this page, also from CP-Algorithms, for explanation and implmentation of Kuhn's algorithm — an algorithm that solves maximum matching in bipartite graphs without talking about flows. As far as I have understood, behind the scenes Kuhn's algorithm works in basically the same way as the flow algorithms I mentioned. It may be faster than the flow algorithms by a constant factor since it is specifically designed for calculating maximum matching, but the time complexity is the same — $$$O(VE)^\dagger$$$.

If you need a faster algorithm for calculating the maximum matching in a bipartite graph, you can use the Hopcroft-Karp-algorithm to do that in $$$O(E\sqrt{V})$$$.

$$$^\dagger$$$ All sites I linked say that the time complexities of the Ford-Fulkerson and Edmonds-Karp-algorithms are slower than $$$O(VE)$$$. This is true in general flow networks, but both of these algorithms work in $$$O(VE)$$$ in flow networks where all capacities are $$$1$$$, as is the case in the maximum matching flow network.

P.S. It may be possible that faster implementations exist elsewhere