My daughter, who is now almost 11, had very similar circumstances at your daughter's age! Definitely work on the acid reflux idea, but consider the following as an "added" help....
Is there any chance, at all, that your daughter is a "grazer"? I'll explain. You know how cows eat small amounts all day long? Could your dd be the same way? You don't want to push the issue, because you'll see that she'll find she gets SO much power when it comes to eating! Instead, think of what she likes - maybe PB&J, chicken, raw carrots, etc. Turn every bit of this into tiny "finger foods," as if planning a party. This stores very well in the fridge.
Now, instead of making a deal at all about this, take a tiny amount of food - for example, instead of one PB&J sandwich, make it two 1/8'ths of a sandwich (one quarter, cut in half - or, better yet, cut with cookie cutters to look like you were just having fun). Put it on the table and say, "Hey, ya hungry?" (Of course, make sure she will be!) She'll be expecting a big meal; she'll see this. Offer her this tiny item. Whether or not she eats it, ignore it. It's a very tiny amount, either way. If she begins eating it, say, "Would you like some [drink - whatever you normally give her, but in a tiny juice glass - you do NOT want her to fill up on milk or whatever!
Whether or not she takes it, an hour later offer her another tiny snack - say, a few cut-up carrots. Always make it sound positive but with a care-free attitude. "Would you like some?" Then leave the plate for a few minutes. Pick it up after a few minutes (unless she's eating). A four year old will NOT starve herself for very long.
My daughter STILL prefers to graize throughout the day. It is much easier for her to eat tiny amounts and keep going than to sit down to a whole meal. LOTS of kids are the same way.
Good luck!
P.S. I just read what Crystal wrote about it might be a texture issue. That's very possible! Read up on that and see what you think.