Breeding During Hot Weather

I'm told rabbits probably won't breed during the Summer months. Never heard of that. Is it true?

Can be.  If they get too hot the males go sterile.  It will last for a couple, three months and then they can breed again.  Mine were OK up to about 90F but they had plenty of cold water and fresh breezes.  If they have access to the ground and shade they can keep cool enough by burrowing a bit.  How they're kept is probably as impactful as temperature.

We raise Florida whites and Californians.  We only have trouble in the winter when it gets below zero for a week.  They won't breed for about a month after that.  Heat never seems to be a problem here.

My fancy & commercial breeds both breed year-round, although there is some summer sterility once the heat kicks in 24-7. Since I have a wool breed, my show stock is in a climate controlled barn; but that just means the temp is around 95° instead of 105° in July & August.
The only real issues I've had breeding year round is the loss of some does and litters due to her kindling during the day and overheating. To minimize that risk I try to get the does bred to kindle at the end of June, that way they aren't carrying a litter or going into labor when the temp is too high. I do remove excess fur from the nest a couple of days after birth, because kits will get too hot and jump out of the box too soon & squeeze through the wire and get lost.
If I do put a doe in with a buck, I wait until night, as neither one tends to cooperate when it's too hot (they just plop down in the cage).

So, I definitely think it's possible to breed year-round, but the methodology may need some modifications to be successful with it.