I believe the answer is something you will need to find out yourself. If I remember correctly there where some data on the product page before but I could not find it again. Maybe it was removed.
The power is generated by a MPM3610 Step-down converter. That has a input range of 4.5 to 21V and a continuous load current of 1.2A. That would suggest you would be find, but there are a few issues.
We have already seen that there is an unusual high ripple in the 3.3V (there were some other posts in this forum). The pcb does not seems to follow the layout guidelines in the datasheet (maybe the Arduino guys are smarter, and the design is OK, but we do not know).
Because the input voltage range is so high, the output current is likely limited by the heat dissipation capability of the MPM3610 package and the pcb. So, the Arduino guys would likely need to provide a graph to show us the values. Many people would probably have difficulties figuring out what the right value would be for them. But if Arduino would write the worst-case value on the webpage, people would probably not buy the board. There is a thermal shutdown feature, so you could simply try how much current you can draw. This will also depend on whether you are using WiFi or BLE at the same time.
The Vout of the MPM3610 at the top pcb layer seems to be very thin. Unless there is an additional power layer inside the pcb it must carry the board current plus your current.
I guess you will need to do a little bit of testing and hope this is a bit helpful anyways.