The Service Academy process is materially the same for any student applying. However, as a recruited athlete, they have an additional voice in the admissions selection discussion. Each team has a number of slots in which they can advocate for on behalf of the student. They have a fixed amount of cards they can play in the conversation and cannot advocate for all students who contact them.
If the student is qualified academically, medically, and from a fitness perspective - they would be offered an appointment. If the candidate is a recruited athlete, but is lacking in one of those areas, prep school could be offered to keep them in the pipeline with a path into the full school.
USCGA does not require a congressional nomination. This is an additional requirement found at the other Service Academies.
As for whether they offer spots, it would typically come in the form of a Letter of Assurance from the school that says "you're in - IF you do these things" and that would include completing the application, completing the medical exams to ensure there are no surprises, and completing the fitness exam with satisfactory results.