Thanks for all the input, guys! A number of good things to check out here. So far I have not done any troubleshooting on it, as this usually happens when I'm in a hurry to move the truck or take it someplace. Hopefully I'll be able to dig deeper into it if/when it happens again.
My pump is the small round type that looks like a metal fuel filter. It is mounted back near the tank and has another metal filter inline between the tank and the pump. As Jerry mentioned, it is very quiet. I assume the filter is not plugged, as the truck runs fine once I get it started. I think the carb is due for a rebuild though, as the accelerator pump is not working the way it should.
I will try giving the pump a bit longer to catch up and see if that makes a difference. Not sure how long I waited last time--maybe it was only a few seconds, although it SEEMED longer than that. Maybe in reality it wasn't.
Thanks again for all the suggestions!