My wife's job places her on long-term contracts (> 6 months) in different parts of the US. The employer currently pays for a permanent car through a popular car rental agency and a permanent room through a popular hotel chain. If her employer is willing to consider alternate companies to rent from, would I be able to create a company that purchases these assets and then rents them to her employer for her to use? Are there any legal or tax problems with this plan?
Technically you can, but it would not be practical.
First legal issues - you'll might have to register yourself as a rental service, not just a private landlord. Companies usually don't rent from private landlords (and from my personal experience in similar cases - they get much better negotiated rates from the chains with which it would be hard to compete).
Then of course the taxes - you'll be paying various income and franchise taxes on something your wife gets either tax free or pays taxes already.
The IRS and the State might not like the arrangement, as well. They might consider it a tax avoidance scam of some kind (which it in fact is), because you'll be receiving money from your wife's employer, but it won't be taxed as salary. From the IRS perspective, it is salary so you might end up being on the hook for all the payroll taxes as well as some penalties and maybe even criminal prosecution.
And last but not least - one time short term rental is not profitable. You'll have to sell that car/apartment when the assignment is over and the overhead will eat up all your earnings.
Get a legal and tax advice from professionals - attorneys and CPA/EAs practising in this field.
Make sure the employer is aware of this arrangement. And if the employer is a government agency, or funded with government money, stay away from such an arrangement altogether.
This type of arrangement is known as self-dealing, where your wife in effect represents both sides of a transaction. That can create a conflict of interest to the detriment of the employer.