Your best bet is to catch Amtrak's Sunset Limited. It's the nearest rail line that goes through San Antonio, dropping you off right at the edge of Downtown in St. Paul Square and next to the Alamodome - this is a fairly short walk to the Alamo itself.
The only thing is that in order to catch it, you would have to take a bus back out west to El Paso, or down south to Alpine, TX. From there, though... You won't even need state rooms. Just coach seats will be enough, and everyone can enjoy the scenic Texas Hill Country on the way in. (The city is called Alpine for a reason!)
https://www.amtrak.com/routes/sunset-limited-train.html
The only thing is that in order to catch it, you would have to take a bus back out west to El Paso, or down south to Alpine, TX. From there, though... You won't even need state rooms. Just coach seats will be enough, and everyone can enjoy the scenic Texas Hill Country on the way in. (The city is called Alpine for a reason!)
https://www.amtrak.com/routes/sunset-limited-train.html