Question

If this is your job you would just include it in your activity level setting.

According to the Compendium of Physical Activities**, "pushing a wheelchair, non-occupational" is a 3.8 MET activity. In the occupational category, "walking, pushing a wheelchair" is a 3.5 MET activity. (They'd be different mainly because of different research studies, I'm guessing - 3.5 and 3.8 are pretty close.)

The MFP exercise database uses METS estimates. A probably roughly 3.5 MET activity that's in the database is "Stationary bike, very light effort". You could log your minutes of wheelchair-pushing as that, or put your minutes into that entry to see how many calories it estimates for you, then create your own custom exercise in MFP for "Pushing Wheelchair" (or whatever you want to call it), using those minutes and that calorie number the first time you log it. When you log that same custom exercise for different minutes in the future, MFP will scale the calorie estimate based on the calories per minute from the first time you used it.

This is a rough way of getting an estimate, but not as crazy-stupid as it may sound, I think.

(Loosely, METS is a research-based way of estimating the amount of energy an activity requires, as a multiple of your resting metabolic rate (RMR). The implication is that two very different activities with the same METS value would burn roughly the same number of calories, per minute. The actual METS values for specific activities come from research. METS is not a perfect way of estimating exercise, works better for some things than others, but the underlying approach is what makes the method I suggested possibly viable to get a good-enough estimate.)

** https://sites.google.com/site/compendiumofphysicalactivities/home?authuser=0