A question

Either way is fine, adding "for" might sound a bit more polite.

Hope it helped.

for is correct, and you can't use at. sorry. For implies that you are doing it for them. Because in the sentence "What time would you like to make it for" you are saying that I'm making the reservation at your request, and want to know what time you would like.

I'm sorry I can't actually tell you why "at" is wrong, I just know that it is. you can use it when you're telling someone you'll be there AT a certain time, but not when you're asking someone else when they would like their reservation

"For" means "earlier than" here. For example:

1. I want to get there at 6pm.
2. I want to get there for 6pm.

In sentence 1, I wish to arrive at 6pm sharp - maybe 5 minutes early or late is okay, but 6pm is the time I want to *arrive*. In sentence 2, I wish to arrive *before* 6pm - I don't care when, but even 1 minute late is unacceptable. I need to *already* be there at 6pm.

A real-life example: "we need to be at the train station for 7:43, so let's try to arrive at 7:30". That is, if we arrive one second after 7:43, we are too late. But, to be safe, we're going to try to get there a bit earlier.

Actually making an appointment "for" a certain time includes the expectation that the customer will arrive by that time or slightly earlier. "Make an appointment for" has the meaning of "be in time for". "at" would be used when stating a time like "Shall we meet at 7 tonight?" In question form, this would be "When shall we meet?" or "At what time shall we meet?" It is a bit difficult to explain.

Actually making an appointment "for" a certain time includes the expectation that the customer will arrive by that time or slightly earlier. "Make an appointment for" has the meaning of "be in time for". "at" would be used when stating a time like "Shall we meet at 7 tonight?" In question form, this would be "When shall we meet?" or "At what time shall we meet?" It is a bit difficult to explain.

You cannot use "at" in this case. The key point is that "it" is a "booking". ie "What time would you like to make your booking for?"

We always say "booking for"+時, "reservation for"+時, "appointment for"+時. It's a set English expression. You can use "booking at"+場所. For example, "Let's make a reservation at McDonalds for 5pm tonight." (though you don't need a reservation there!)