Which turbotax version should I use if I need to need to report moving expenses on a Form 3903? Does the free edition cover this?

Moving expenses are not deductible on a federal tax return unless you are active military on Permanent Change of Station orders.

no.

You can deduct moving expenses only if you are a Member of the Armed Forces on active duty and, due to a
military order, you, your spouse, or your dependents move because of a permanent change of station.
some states may allow it.