Phrase for pooling funds with others for a purchase

In American English, the most common phrase would be to chip in, as in I would be happy to chip in for a cake:

to give some money, esp. when several people are giving money to pay for something together

An alternative is to pitch in, though in my experience this connotes a contribution of labor rather than capital. Both chip in and pitch in can also mean to make a small (and perhaps unnecessary) comment, so don't be surprised if a request for others to chip in yields more quips than coins.

Speaking for myself, I would offer to go in on a cake casually, meaning to join or

to be involved in an activity that involves others

but which my peers and I often use in the sense of investing in or purchasing something together, like a pizza or a vacation condo.

More formally, I would offer to contribute towards a cake. That is, I don't intend to purchase the entire thing, in which case I would contribute a cake, but rather I want

to provide money or support to help another person, company, or organization to achieve its goal

I hear and use "happy to chip in" more often but given the context of your reply, I'm sure it would be understood - just not common usage, that I'm aware of.

I hear pool used in your situation all the time. We will pool resources for each division for this operations project. The other parents should understand you.