Not getting much feedback here I see. Maybe this post will bump the subject into someones vision. San Jose has a fair bit of competition for what might be available. Construction folks, homeless, and just your regular run of the mill grass roots need a reasonable place to live kind of folks all tend to suck up space. So you may need to vary your approach to find a solution. Commuting is a pain, but obviously, it expands your options.
The Gilroy Garlic RV Park is a bit of a drive from San Jose, but a favorite for those folks who chose it as their temporary home. The manager controls the park and residents seem to like that approach. Note, you will find RVs here and not park model trailers. This location may be too far for you, but is the type of place you likely need to find.
When I commuted to San Jose, it looked like some side streets behind industrial buildings might be a good temp solution, but most of those no longer have unrestricted overnight parking. Local security keeps folks moving as most are considered a problem. Local Walmart gets a bit of pressure and from time to time you see folks find unusual spots that work until someone has a problem and objects.
There is not much police protection in SJ as the force migrated out due to pension reform. They are rebuilding and it will get better. But for now, finding a close and supportive community is likely a very good objective.
Good luck with your research. Sorry I don't have the perfect solution. Sometimes it's who you know that gets you what you want/need. Your new co-workers may have just the fix.