To be a bit pedantic of heraldic terminology, you're talking about a coat of arms. The crest is really just the top bit above the helmet (if any).

Coats of arms belonged to families or sometimes individuals. It depends on the country; different countries had different traditions in almost every respect. First there are different traditions in which groups of people had them; the nobility almost always did but in some countries it was quite common among other social classes as well, and even sticking with nobility there were also big differences between in how many nobles there were relative the population; some countries had very large groups of petty nobility, some had few. How titles and coats of arms were inherited varied as well.

But in any case, most people of European descent simply don't come from families with coats of arms. Meaning they have no direct ancestor who had one. So if you don't know what your coat of arms is, it's far more likely that your family never had one in the first place rather than it was forgotten. (although nothing says you can't invent one for yourself)

Coats of arms aren't granted for a surname; they belong to a family and sometimes they've belonged to a single individual of it. Unfortunately there's a lot of unscrupulous web sites out there taking advantage of these exact misconceptions and which will happily tell you what the 'crest for your family name' is in order to sell you stuff with it.

That doesn't mean they're 'illegitimate' in the sense that they're made up; in most cases I think they're not; they've digitized old armorials (books of coats-of-arms). Rather they leave out context and misrepresent things, the "crest for Smith" they might present may have been used by one particular John Smith who died childless 300 years ago. So it doesn't really 'belong' to anyone today.

Depending on which jurisdiction you're in you may or not be free to use anyone's coat of arms whether you're related to them or not; some countries have restrictions akin to trademark law on that, but in most (globally speaking) it's entirely unregulated.

But if you don't know who first used that coat of arms and how you're related to that person, what's the point? You'd just - in effect- be claiming to be related to someone, or to being a member of some family, without actually knowing whether you were or not.