Switching web hosts will not affect your web site ranking in any way.
There is only one thing that you should take into consideration. When you change your domain's DNS by pointing it from one server to another, there is a propagation period before the new changes take effect. You should not bring the old website down until you are 100% sure that search engine crawlers are already indexing your website on the new server. A safe course of action would be to:
1. Migrate your web site along with all the content and data to the new server.
2. Change the primary domain name's DNS to point to the new server.
3. Observe the new server log for visits from search engine crawler bots.
4. When you notice that the search engine crawlers are indexing the web site on the new server, it is safe to bring the old one down.