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To find Alumni Clubs, I recommend looking within a few hours driving radius of MIT and seeing what alumni clubs are active. Email them and just ask if they'd be willing to host you guys and your performance. Every Club you deal with will be different in what they want or can provide, but having a support in the location you are going to makes planning easier and ensures you have an audience when you get there.
When to Tour?
Fall Semester:
In my time, we never toured during the Fall semester since the on-campus show dates were right near Thanksgiving and a weekend trip (or Tour in general) didn't seem feasible.
Spring Semester:
In 2019, we performed during the second weekend of spring break (leave MIT the Friday of spring break, return Sunday, the day before classes begin). This timing worked well enough, but it resulted in what felt like a very short spring break. Now that the last performance of the spring semester is the Friday right before spring break, an alternative that we voted on for the attempted Tour of Spring 2020 was leaving campus that first Saturday of spring break and then returning to MIT by Monday, thus allowing people approximately 5 uninterrupted days after Tours to return home or relax on campus. Also coordinate with the alumni/performance location on this to match their needs, but at the end of the day, it matters more that people are available and willing to Tour the selected weekend rather than where exactly you are in performing.
Where to Stay:
In 2019, we stayed with the alumni of the Hartford Alumni Club. The overwhelming response to this was that it was unnecessarily stressful, and it resulted in Tours being quite unfun since everyone was split up. I do not recommend doing this again. Instead, try and find reasonable (read: cheap) hotels near where you are performing. Alumni can likely help fund this as well. Once you have an accurate headcount of how many people are going (actors, production staff, extra hands if needed), book the rooms, and then record from those going rooming preferences. Some things, like how comfortable someone is rooming with a certain gender or individual, are really important to take into account. Also, it's generally good practice to keep couples separate. If you receive push back from Alumni who want you to stay with them, feel free to cite MIT Policy, which does say that the Institute does not like students staying in the homes of Alumni.