I bought a minivan to help us run our nonprofit more effectively. Can the purchase of the minivan be used as a tax deduction?

Almost all of this question is going to raise red flags with the IRS.  You need legal and tax advice on running a non-profit.  I will assume that the organization is registered with the IRS as an approved tax exempt organization eligible to receive donations.  If not, you are in even bigger trouble.

First, in the case of any donation of a vehicle to a non-profit, you can't take a tax deduction unless you donate the entire vehicle--title and all.  You can never take a deduction for donation of a partial interest in something or the use of something.  You would have to give the vehicle title to the organization and the organization must issue the donor a written receipt and a form 1098-C.  If the charity owns the vehicle, then any personal miles driven by you or your wife as principles of the organization must be treated as taxable compensation.

If you retain title to the vehicle, you can claim mileage driven on behalf of the charity as a donation at the flat rate of 14 cents per mile.

You can never deduct a donation as an unreimbursed employee expense, including other startup items.  They might be charitable donations if the organization has an approved EO registration.  Your wife might be able to deduct unreimbursed employee expenses if she was an employee.  But you said she takes no salary yet so there is no employee income to deduct expenses against.  Startup costs could be reimbursed tax free from donations under an acceptable reimbursement plan, but see below.

The fact that you are principles of the charity raises issues of personal inurement and illegal personal benefit.  It is not illegal to reimburse yourself from charity funds, but it is dangerous if you are the person who controls the charity funds and decides which expenses are legitimate.  The IRS, or a disgruntled donor, could accuse you of personally enriching yourselves by paying for personal expenses from charity funds, or by giving yourselves tax donation receipts for tax deductions for things that are ineligible.  It would be equally risky if, for example, your other job is a catering business and you decide to use charity funds to pay for a fundraising dinner where you are the caterer and get paid for that service.  You pretty much need a firewall between personal funds and charity funds and ideally, you need someone else at the charity (another officer) to co-sign any arrangements that you might appear to personally benefit from, so that you have backup that the arrangement was to the charity's benefit.

Please get legal advice before you file your personal tax returns.