Budget does not let you pick your own vehicle. You are assigned one, or they will assign you one at the counter depending on availability.
As for
Alamo/National allowing you to pick a car, the process is fairly straight forward but different depending on your status. In general, you will need to go to the counter unless you sign up for their frequent renter program (which is generally free).
If signed up and preregistered, you would normally go to the aisle of the car type your rented (mid sized, SUV, Minivans, economy, etc), pick a car you like, drive it to the gate, present your documents (frequent renter card, drivers license, credit card), then drive away once you have the contract.
National has two options for their frequent renters: rows of cars in the class you rented and the Emerald Aisle. If you rented a midsized car, you can choose from the selection on the Emerald Aisle instead of the mid-sized car aisle if you want. There are generally a variety of cars there from mid sized, full sized, small SUVs, pick ups, the occasional mini van, and others. However, some people just like their mid-sized car, so they would pick from the mid-sized section. No class other than mid-sized rentals can pick from the Emerald Aisle.
Now as for MCO, not sure on their specific rules, so we'll need someone more versed at this specific location to chime in. However, the above is how Alamo/National operate, in general. The nuance is important, if you've not done it before.
