Welcome Meg! This is a great group of girls and we will try and answer questions if/when we can.
I am by no means an expert, but I would start off by checking with the University first about rules for Greek housing. They vary from school to school--some Greek houses are universtiy owned, some are chapter owned, some are rented--it just depends on the individual chapter and school.
Imsohappythatiama can give a better answer, but I believe that, where possible, the chapter house is owned by the chapter--through the House Board Corporation.
It is not an easy feat to get a chapter house--even for a smaller chapter. There are so many different hoops that need to be jumped through before the process can even start. It is a lengthy process that is often very very expensive and time-consuming. There needs to be a strong alumnae base near to act as House Board and make sure all the details are followed correctly. I would do a search on GC for building new chapter houses to gain some more insight on the process.
What is your housing situation like now? Do you live in residence halls and have a chapter room somewhere? Do you mainly live off-campus? What are the issues you feel you're struggling with? Perhaps someone who didn't have a chapter house yet still was a member of a cohesive chapter has some suggestions that may help. And always, you can get in touch with your PDC, RDC or HQ with any concerns you may have about your chapter.
Good luck!!