You should not be concened about asking for a required-by date - you're just collecting information in order to serve your customer better.
You do need to be careful about agreeing to a date, and you can't do this until you know -
- how long the job will take, and
- when you will have that time available.
Once you have those two bits of data, you can safely commit to a specific date.
So, when you are asking the customer when they want it by, you are either just collecting information, and not making any commitment (make that clear) or you'll be able to tell the customer whether you can meet the date or not.
If you can't honestly commit to the date the customer wants, then some negotiation needs to happen, about what tasks you do when. Whoever does that, they need to understand the overall priorities.
Also, sorry, but I think you're best simply asking when it's required, rather than asking for 'the latest date you need this' or whatever. Just be straight and open.