There are many magic items with charges that replenish at dawn. However, there are places where there is no dawn, like in the Feywild.
I'd think this way of phrasing it was only meant to make sure it restored once per day. But you could argue that since it's specifically phrased to be at dawn, it doesn't restore in the Feywild.
Do they actually replenish at some point if there's no dawn, or will they not replenish at all?
This is addressed in Xanathar's Guide to Everything, page 144:
Some magic items can be used a limited number of times but are recharged by the arrival of dawn. What if you’re on a plane of existence that lacks anything resembling dawn? The DM should choose a time every 24 hours when such magic items recharge on that plane of existence.
Even on a world that experiences dawn each day, the DM is free to choose a different time — perhaps noon, sunset, or midnight — when certain magic items recharge.
