What size mesh do you recommend for me to use for this logo printing on vinyl?

If you're printing on a non porous material like a vinyl product with an air dry vinyl ink that is on the thinner side then use a higher mesh count like 200 and a 80 durometer squeegee. If the ink is very thin use a 230 mesh. Since the substrate is smooth and does not absorb ink we can print a lessor volume of ink which will help make it easier to achieve and keep crisp fine line details. Here's a sign video that may be of use to you in that it shows printing of an air dry Coroplast ink onto corrugated plastic sign sheets. It should be a similar process so don't forget to back flood.