I'll echo the suggestion that you dig deep and have the will to overcome social anxieties and get a gym membership if you can. My Life After Fat™ has gone through three distinct phases:
1) Improving my diet and starting to run for the first time in a decade.
2) Buying a standard barbell set with two 50lbs weights so I could DEADLIFT.
3) Buying a gym membership.
Phase three has seen my best body recomposition results and my best strength gains. People love to help the newbie and are always happy to answer any questions and no one ever gave me grief when I could only bench who-knows-how-little back then and had never done a proper back squat. I was lucky to receive a good chunk of old-school knowledge from an Olympic weightlifter that was still squatting 500lbs when he was in his early 50's ten years before I met him. He was still always trying to improve himself every day, even in his early 60's. Of all the people I met he was a great role model and an example of long-term strength progression and general health.
If you insist on staying home for a bit at first, your progress will be slower but it isn't impossible. It's just something we'll have to work with. From my experience and the knowledge I've gleaned, the least you can add to your equipment list is a good barbell set. That way you could deadlift and press, the bare minimum any Joe or Jane in the street should be doing in my less than humble opinion. It's what I did, more or less, but with the help of people in this thread you could be much more efficient about it than I was and make some decent gains in the months before you get a gym membership.
I'm not a fan of P90x. When adding size to your upper body is a goal, P90x isn't entirely aligned with that in mind. Not to say you wouldn't see improvement, but there are better ways out there and I like to separate cardiovascular exercise and resistance training where ever possible.
GAF being GAF, I'd be surprised if two-thirds of the people in this thread didn't start off like HarryHengst or myself. It's intimidating at first, and it's going to make you uncomfortable, but I've found it to be an enlightening experience. Even though I'm not going to renew my membership when it's up, it has been the best investment in myself I've made RE: fitness.
Keep us updated on where you choose to go and what program you select. Always feel welcomed to ask questions if you can't readily find the answer.