Ironically, it may be easier to get both loans at the same time.
If you get the first one and then wait a month or two...........that first one eats into your debt to income ratio and may cause you to be overextended in the eyes of the credit company.
If you pull the trigger on both on about the same day, you may be able to get both because the system is not really set up to deal with 2 loans simultaneously.
There are several variables that can work for and against you.............your credit score, your monthly income, your exisitng loans.
When you shop for cars and fill out a credit app, the dealer will run a credit check---and it is not unusual for a customer to have 3, 4, 5 inquiries while shopping. The FICO model is set up to basically ignore these as it knows it is traditional for multiple reports to be pulled.
Here is my biggest warning..............make sure your loan(s) have all the work finalized before you pull the car off the lot. Do not take the car home if they offer to let you have it as a courtesy because they "are confident that the deal will go through". This is an area where dealers and customers have had problems in the past. If you have a trade in this is even more important as your trade may be wholesaled the next day and a few days later the loan for your new car comes back declined..............Oops!
I have not been involved directly in this side of the business for many years, so I hope someone from the credit industry chimes in. The industry may be a little quicker than it was a few years ago.