Do we need a Board vote to change management companies?

Almost always yes, and doing it properly is what makes the decision durable if anyone challenges it later. A management agreement is a material contract, and in nearly every state and set of bylaws that is a Board action, not a presidential one.

What the record should show: that the Board considered the matter at a duly noticed meeting with a quorum; a motion identifying the company, the term and the fee; the vote count; and the officer authorized to execute. If your act permits contract discussion in executive session, deliberate there and take the vote in open session — the deliberation stays candid and the decision stays on the record.

Homeowner approval is generally not required for a management contract. Check your declaration anyway; a small number impose spending thresholds that require a membership vote, and it is a cheap thing to confirm.

Then attach the comparison to the minutes. A Board that recorded which companies it invited, which responded, what each proposed and why it chose one has a defensible record. A Board with a motion and no supporting material has a decision that is entirely dependent on people's memories.