Is it legal to scan a website for vulnerabilities?
Reading what a site sends to every visitor (headers, pages, scripts) is what browsers do. Sending attack payloads to a site you don't own or have written permission for is not, in most countries. Passive checks on anyone's site, active scans only on your own.
Can a free scan find every vulnerability?
No. Outside scans can't see business logic, access between users, or most of what sits behind a login. They find the common, cheap-to-exploit problems, which are most of what automated attackers use.
Which free tool should I run first?
One that checks your browser code for keys and your database rules, because those leak data directly. Header grades matter, but a missing header is rarely how a small app gets breached.
How often should I scan?
After every release that changes sign-in, payments or database access, and at least monthly otherwise.