What do you call someone who makes a request?

If you are not limited in space, you could use

Do you really want to send an email to the person who made the request?

Otherwise, requester is an acceptable English word to express this same thing.

For this context, might you be able to get away with user, customer, or client?

Depending on your context, you can use Requester or Requestor

In diplomatic terms, the word is a demandeur.

For this context, might you be able to get away with inquirer.

Solicitant – a generic name for a person who makes a request. Rarely used in English.

In French one would say "suppliant" translated supplicant.

Sly, why, please, would it be difficult to use, for instance, Do you want to send an e-mail to Tommy Atkins? What problem might that cause? Equally, why would Kit’s Do you really want to send an email to the person who made the request not work?

Did you really mean I’m designing a software, please? Did you really mean a request detail window? In many places those questions might be irrelevant nit-picking and here, they matter. Here, it matters exactly why a field needs to be named Requested By and if it does, why the message needs to use the field name, specifically? Modern English just doesn’t get specific in areas like that.

Please consider who might be making this request? Could it be a colleague, a supplier, a customer or potential customer? In any case, does it need to be a named individual or might it just as easily be an organisation?

If you must make so many different aspects of your SW design dependant on what will necessarily be a problematic noun, why not use questor, as at ? That might sound a little odd but no other word could fill your bill better, nor anything like as well.

Where I work we use the term "caller" on electronic forms for someone who makes a request, as someone who "calls" for a request.