Not based on the company, but the speed of the server can have an effect. So if you have one account with the monster and one with daddy and they are both shared hosting and under a heavy load your site will load slower.
If your site loads slow, it could and likely will not rank as high. This is not that critical in smaller niches, but if you have a lot of traffic and major keywords it could be crucial in a heavy comp environment.
You can always upgrade your server account to dedicated server or a virtual dedicated if you have over say...10k unique per day.
If you are only targeting long tails and getting 1 k visitors a day I don't think the added expense of upgrading.
I keep my sites on shared until I get top rankings and then I upgrade to a better account plan.