What you describe is called a "partnership" ("General Partnership", more precise).
Partnership are unincorporated associations of people with a common goal in mind. Every partner shares the same responsibility and obligations, and the duties and authorizations to act on behalf of the partnership should be written down and signed by all the partners in a contract, which is called "Operating Agreement".
With that in place, you (if you're given the authority by the partners) can open a bank account on behalf of the partnership, and allow other partners access to it (with or without signature authority, per the operating agreement).