Sometimes, but not typically. The film’s producer makes that decision.
Many film productions outsource it to a motion graphics company like Scarlet Letters, which claims to be the number one company in the world specializing in end titles. The company might receive a credit for that work, but the specific person who types the names into the end credits won’t usually receive their own individual credit.
That’s because they’re not doing a ton of work—the production coordinators are typically responsible for keeping accurate crew lists, and the person “writing" the credits simply copies those lists into a piece of software.
So, how could you tell whether the credit-writer is in the credits? The credit for the...uh, credits might be listed as “Titles by…" or “Credits by…,” but if the company that created the credits had a more significant role in the production (for instance, if they did other graphic artistry work on the film), the credit-for-credits might not appear—after all, the company has already gotten their due.
We should note that there aren’t many established rules for listing credits, although unions like the Producers Guild of America and the Directors Guild of America have some requirements for studios that use their members' services. When space is at a premium, the person who actually “wrote” the credits won’t be in the credits—and their production company’s name may or may not appear, depending on the producer’s preferences and contractual obligations.