is it possible to call the animal shelter to see if you can pick up an animal when you have time

Give them a call. They are helpful. They do seem to have an adoption reserve system.

Yea 100% you can reserve animals like the object they are

Absolutely not! Everything is ON THEIR TIME

I would say that they do not make reservations and also you would have to adhere to their regular business schedule.

Normally you have to take it unless is not yours.