Does Budget Forward need to connect to my bank account?

No, in fact it was specifically designed so that you do not need to connect it to your bank account.

Many products require or encourage access to your bank account details for the purposes of analysing past spend and keeping your net worth up to date. Budget Forward, by design, helps you plan for the future, so past spend, while interesting, is not required to formulate a plan and simulate your future results. This only leaves the current balance which, if out of sync with the real world, can easily be updated to reflect reality without having to share your bank account details.