Admissions to US med schools for US citizens is primarily a matter of grades and test scores, although other factors matter. If you have the grades and test scores, and have otherwise checked the necessary boxes (good letters of rec, related volunteering, etc…) where you go to school as an undergrad will not matter greatly. Howard is fine. Yes, there’s a bump for a ‘prestige’ undergrad, but not a lot. The data you want is here:
Just to give you a sense of the cost, a state school is around $160,000 for 4 years. A private is around $240,000. Good that you have family members who are willing to pay. One less hurdle on a long a difficult road.