Number of bounded minimum vertex covers
The number of inclusion-minimal vertex covers of size at most $k$ is at most $2^k$: They can be enumerated by a search tree algorithm that initially has a budget of $k$, then branches as long as the graph contains an edge $\{u,v\}$ into the two cases to add either $u$ or $v$ to the vertex cover. Each branch can be solved recursively and the parameter $k$ is reduced by one in each case.

The number of minimal vertex covers of size at most $k$ obviously bounds the number of minimum vertex covers of size at most $k$ as well. 

Further information can be found in  

Henning Fernau:
On Parameterized Enumeration. COCOON 2002: 564-573 dx.doi.org/10.1007/3-540-45655-4_60

and

Peter Damaschke:
Parameterized enumeration, transversals, and imperfect phylogeny reconstruction. Theor. Comput. Sci. 351(3): 337-350 (2006)
http://dx.doi.org/10.1016/j.tcs.2005.10.004