For sales tax purposes, any vehicle that you are trading in can be used to offset the taxable amount of the purchase. That means your brother-in-law can give the dealership the car as part of the trade-in value to reduce your purchase price and the tax owed on that vehicle. However, your brother-in-law may be liable for gift tax if the value of the car is above $19,000, otherwise there is no issue with him gifting the vehicle through the trade-in. Illinois has removed the $10,000 cap on trade-in deduction as of 2022. More information on the trade-in deduction can be found here and here.

The play to reduce the sales tax is to have the dealer take both in on trade against the vehicle that you are purchasing. If you sell either/both cars, you owe sales tax on the full purchase price of the vehicle. However, there may be some hoops to just through with the title on the brother-in-law's mother's car if it is still titled in her name. BIL may need to transfer title first to their name, then do the exchange. Do not transfer to your name or you would owe use tax on the transfer. If BIL is transferring from his parent to BIL, then there should be no tax due because it is a sale/transfer from parent to child.