If the stipend from work has been received for the past 2 years and is expected to continue, the details would need to be reviewed, but those are normally the requirements for extra income like that to be able to qualify. It is treated as extra income rather than offsetting your rental expenses (i.e. your income goes from $3,333/mo to $4k/mo, etc.).
Keep in mind that some lenders do not permit a first time homebuyer (someone who hasn't owned a home in the past 3 years) to purchase a 2nd home or investment property, so if that is you then it'd be an important question to ask upfront (don't expect lenders to volunteer that information, it may not be "caught" until further along in the process).
If you could qualify for a $90-120k home with 10%+ down & $40k of income would still depend on: your stipend amount, what you pay in rent, and how much your other consumer debt payments are (car, credit cards, student loans, etc.). You'll want to keep the total payments, including the new mortgage, to no more than about 41% of your gross monthly income.
If you need assistance figuring out a plan early next year I'd be happy to help.
You didn't mention down payment. I believe with enough down payment this wouldn't be an issue. It will depend in large part how the lender qualifies you. This will not be your Primary Residence, so you'll need to qualify as either a vacation home or investment property.
Your only reliable answer will come when you sit down with a lender who can do multistate loans or an AZ lender. It can't hurt to investigate further, best of luck.