Can the 737 Max autopilot be engaged when some sensors have failed?
The following applies to the Boeing 737 NG series, but probably there are no large differences for the MAX series.

The FCOMv2 (4.20.2 Automatic Flight - System Description) mentions only the following systems in the Autopilot Engagement Criteria and the Autopilot Disengagement sections:


  The A/P automatically disengages when any of the following occurs:
  
  
  [...]
  either left or right IRS system failure or FAULT light illuminated
  loss of electrical power or a sensor input which prevents proper operation
  of the engaged A/P and mode
  loss of respective hydraulic system pressure.
  
  
  Note: Loss of the system A engine-driven hydraulic pump, and a heavy demand
  on system A, may cause A/P A to disengage.


The sensor input is probably what you are asking about because this would affect the instruments as well. In general, if a system only affects one side, the other autopilot should still work. One notable exception would be the IRS, where any IRS failure causes the autopilot to disconnect.



Edit:

The two autopilot systems on the 737 are called A and B, and they are not necessarily linked to Captain and First Officer. Both the Captain and the First Officer can press both engagement buttons on the MCP (marked with 1 in the image from the FCOM below).