(Assuming the jurisdiction is the US.)
I was wondering whether there are any limitations on using photos of
private individuals on websites.
is the least of your potential problems. Photos of the public taken in public are mostly legal to use and publish, and you own the copyright on the photo, and generally don't need a model release.
But your plan of posting the photo with accompanying information about (alleged) fraud:
There's a fraud conducting business in my state and I want to create a
website that warns others of his fraudulent practices.
is potentially legally problematic.
As phoog indicates in his comment, you need to be aware of defamation, both at the federal level and among states, as some have criminalized defamation (Wikipedia).
Libel is the publication of provably false facts by one person about another person. For a full outline of defamation (libel and slander, and including private and public figures), see Libel and Slander | Nolo.com.
What you want to do is publish "facts" about this alleged fraud on the website with a photo of the individual, linking the fraud allegations to them. That is potentially libelous.
The facts of the fraud may be provably true, or they may be provably false; that remains to be seen. But the facts don't matter when considering what actions the person can take against you if you publish such information on your website.
If this business and the individual(s) feel they are not committing fraud, they can sue you for libel in civil court, alleging that you have damaged their reputation and impacted their business by publishing those facts on your website with the photo identifying the person and their business.
If they take legal action, and you can prove the business and the individual(s) are committing fraud with provable facts - hard evidence of fraud, such as legal documents and court judgments - than you should (no guarantees) be able to successfully defend yourself in a libel suit brought against you by that person.
Even if the business and the individual(s) are aware of facts that prove their fraud, and know they will probably not prevail in court, they can still take you to court, and it will cost you whatever time and money it takes to defend yourself.
My responses above concern what legal actions the alleged could take against you if you published the website with the photo and information about the alleged fraud.
No one here is advising you to put up the website with the photo and the "facts" as you see them, even if you have hard proof of the facts of the fraud; you should find legal representation before taking any action with the website. And, no one here is advising you to open any legal action against the person; that's your choice in terms of determining your case and if you can show actual harm that was caused by the alleged fraud by the individual, and you should find legal representation before taking any action.