By design axe eyes are typically softer than the bit or bits of an axe. This is done so the eye is less likely to crack. As a result they can be dented or expanded relatively easily. But they can also be repaired relatively easily.
There is a basic principle in blacksmithing. Steel moves easiest where it is the hottest. You can use this principle to your advantage. First file off any burrs on the top and bottom of the eye. Half-round files are great for this with round files used for the ends of the eye. Once the burrs are gone tie wet rags around the bit(s). This protects the bits from losing their temper. Then with a simple propane torch like a Bernzomatic, heat the most collapsed side of the eye for a couple minutes. It won't reach red hot but it will soften some. If you lack a vise you can hold the axe with a large pair of Channellocks or vise grips during heating. You will open the collapsed eye with a drift, a tapered piece of steel that can be driven into the eye from each end. If you lack a suitable drift you can make do with a cold chisel from the hardware store. If you lack an anvil you can do the work on a hardwood round.
Once one side of the eye has been pushed out quench it and do the other side next. Since you never got the eye up to austenite phase (non-magnetic, ~ red hot) your quench will not make the eye hard. That's a good thing.
You can use the same process to close an eye that has been expanded by using the axe as a sledge or wedge. Here you'll hammer the hot side closed with no need for a drift.
Recall that the eye deformed cold. You won't need to reach forging temperature to correct it. I've fixed many axes this way.