Without a location tag, I'm going to assume the United States. According to Section 3-401(b) of the Uniform Commercial Code (my emphasis):
A signature may be made (i) manually or by means of a device or machine, and (ii) by the use of any name, including a trade or assumed name, or by a word, mark, or symbol executed or adopted by a person with present intention to authenticate a writing.
Based on a plain reading of this, you can use a stamp in place of a manual signature. And part (ii) of that same section makes it clear that you can use "MrPeanut, LLC", instead of "John A. Smith", under the "a trade or assumed name" phrase.
Ultimately, you want to check with your bank. With the banks that I've dealt with, I've had to notify them of my intent to use of a signature stamp to sign checks. (They make a note on the signature card to indicate that a stamped signature is acceptable.) But I've not had to get approval to use a stamp to endorse the back of a check for depositing. They did, however, suggest that I include the account number in the stamp. (I don't know what additional security this provides.) In my case, the endorsement stamp looks like this:
For Deposit Only
My Full Name
(This was before mobile deposits were a thing. Nowadays, I'd make it start with "For mobile deposit only at MyBank".)