Can I request a removal of my account and personal details from a retail outlet database?

Sure you can.
How depends on the company and where you are located.

To my knowledge it's pretty unlikely you'll get much satisfaction there, if you mean what I think you mean. While you obviously could ask, it's likely there isn't a normal procedure for that.

What is the root thing they're doing that might be bothering you? Is it possibly junk mail and/or unsolicited telephone calls, etc. Please provide more information. Thanks.