Can my For Profit Business have a Non-For-Profit Mission/Project

Ask a suitable lawyer and CPA. To do this legally with two entities it requires a lot. 

There are other alternatives that may satisfy your desired outcome, such as a b corp. 

Look into b corps

"For example, I would love to collect school supply donations and deliver it to the struggles school- can I legally collect donations?"

Speaking gnerally, yes, you can legally collect donations through a for-profit business. However, since you do not have tax exempt status the donations would be taxable income to you. When you make the donations to the school, you can apply the IRS rules for them to your business or personal tax return (depending upon what kind of entity you have). Since you are not a 501(c)(3), the customers who make the donations cannot deduct them on their tax returns.

Note that the above paragraph contains general rules. Many states have specific regulations regarding fund raising. Be careful to review those laws before starting.

If you're doing less than $2.0M in annual revenue, probably worth just partnering with a local non profit where you know the Executive Director and BOD and you feel comfortable to give/volunteer. If you're larger than that, talk to your attorney and accountant about benefit corporations and/or social purpose corporations.