erica,
Thanks for the clarification. That makes it easier as I'm quite familiar with Monterosso. The only restaurant I can think of that might require reservations is probably the somewhat upscale Miky restaurant (I'm not sure, I've never dined there) or possibly Hotel Porto Roca. The Cantina di Miky (close to the train station and a big favourite with many here) likely doesn't take reservations (AFAIK), nor do many of the others. As someone else mentioned, you can often avoid crowds by dining early (or at least get seated before the hordes arrive).
If you need more information on restaurants there, send me a PM.