Is it possible to have rounded (fillet) edges on a PCB?
Yes, it's certainly possible with the machinery they typically have.
They would have to use a solid carbide rounding end mill, probably with 1/8&quot; shank only, and flip the board for a second operation. And the boards cannot be stacked for those operations. There might be complications with holding the board in the panel until the last second (maybe another operation).

Sounds expensive and they may not want to do it for a small order. Additional programming and they would have to source the tooling and maintain it.
As one data point, we had one offshore supplier decline to supply countersunk holes (for flat-head screws) on an order of a couple thousand USD. And that was a simpler operation (all from one side). And it could have been done easily as a manual secondary operation (drill press with a stop set).