It is possible to start working for the USDA straight out of school.
Britzen and I were both in the Saul T Wilson scholarship program that allows us to go directly into the APHIS workforce after successfully completing vet school. I had a classmate who had a similar scholarship from FSIS (Adal A Malak). FSIS is honestly the easier route to gain access to. They usually have a wide number of vacancies, and they're willing to work to accommodate applicants. I had another classmates who got hired for FSIS during 4th year. APHIS doesn't go out of their way to make applying to jobs easy for graduating seniors. All of their job postings I see require you to have your conferred DVM degree before the job posting closes. They also don't time job announcements to coincide with graduations.
The pay is..... underwhelming. You aren't going to get rich working for the government. I had a number of offers for small animal GP jobs that dwarfed my federal starting salary. The federal system always starts you at the lowest grade they're allowed to, which if you only have a DVM is usually a GS11. That's around 65k annual salary, it can be slightly higher in higher cost of living areas. After a year as an 11 you can get promoted to a 12 in most position, and that's around 77k. For every year you stay at the same level, you can go up 'steps'. These are smaller raises that occur when you're staying at the same grade level.