Can I run only retargeting ads if my ad budget is very small?

Yes. Facebook ads already have retargeting built in, because Facebook knows who visited your site and who is likely to return. With a very low budget you can run a retargeting-only campaign, so people who visited your site see your ads afterward, which makes your business seem larger than life. You need visitors first, for example from social media posts or from people finding your site through search and AI answers (AEO), so retargeting complements those.

A retargeting budget is much lower than a prospecting budget, perhaps $5 a day or less until you have good traffic. Check the math: if your CPM is $100, $10 buys 100 views, so ask whether you have 100 people to show the ad to, or whether one person would see it 100 times in a day, which is too much.

Once the Facebook pixel is on a new site, do not send friends and family to it. Facebook has no data on a new site and will treat those visitors as your ideal customer.