I have a flash card that reads "Ithim arán le haigh lón". I believe the meaning is I eat bread with lunch, but I don't exactly know the meaning of "haigh", and I can't find it in the dictionary. Maybe I have the sentence or meaning wrong???

Ithim arán le haghaidh lóin = I eat bread for lunch

le haghaidh = for
le = with
aghaidh = face (h is just prefixed because of le)
For unknown reasons "with face" is used for "for" in Irish.

So, I'm assuming that this would actually be with lunch: "Ithim arán le lón". Also, would "Ithim arán le don lón" be equivalent to my original corrected sentence meaning for lunch?

Yes, with lunch = le lón

le don makes no sense.

le do lón = with your lunch
leis an lón = with the lunch
don lón = to/for the lunch
do lón = to/for lunch (same spelling: your lunch)

Possibly you can substitute do(n) for le haghaidh here (I'd think rather don )
So: Ithim arán don lón.

My mistake. I meant to write Ithim arán don lón, but I did a cut and paste and forgot to erase the "le". Thank you.