Yes, computers can handle base 20. The IBM 1401 business computer (1959) could optionally support pounds/shillings/pence in hardware, where there were 20 shillings in a pound and 12 pence in a shilling. (Britain switched to decimal currency in 1971 for reasons that should be clear.) It could add, subtract, multiply, divide, and format pounds/shillings/pence as well as its normal base-10 arithmetic. So it was doing base-12 and base-20 arithmetic.
I should point out that this was implemented in hardware with transistors (lots of germanium transistors), not microcode or software. In other words, the three fundamental hardware datatypes of the IBM 1401 were arbitrary-length decimal numbers, arbitrary-length strings, and pounds/shillings/pence. Of course there were two conflicting standards on how to represent pounds/shillings/pence, so there was a knob on the computer's front panel to select the standard.
(This isn't directly related to the Inuit base-20, but I'm sure IBM would have supported Inuit base-20 if customers would pay for it.)
I should point out that this was implemented in hardware with transistors (lots of germanium transistors), not microcode or software. In other words, the three fundamental hardware datatypes of the IBM 1401 were arbitrary-length decimal numbers, arbitrary-length strings, and pounds/shillings/pence. Of course there were two conflicting standards on how to represent pounds/shillings/pence, so there was a knob on the computer's front panel to select the standard.
(This isn't directly related to the Inuit base-20, but I'm sure IBM would have supported Inuit base-20 if customers would pay for it.)