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They have never done that for me and I have purchased multiple firearms from Buds. You may be able to request that info after it has shipped at 1-859-368-0371. But I'm not sure its possible.
You will actually get the serial number as soon as it arrives at your FFL. I am not sure if they can give it before because they deal with so much inventory and it would be tough to guarantee. You can always call 859-368-0371 or use the Chat Now function (they are awesome)
You will actually get the serial number as soon as you inspect the firearm and take possession from your FFL
Probably the best bet to get the s/n as soon as possible it to track the delivery and call your FFL once it arrives. The shipping / layaway folks at Bud's will have the information but getting to talk to the right person at the right time may not be the easiest thing to do.