The IAP2 framework for public participation is considered a best practice by many and applies to all government decision making that affects that public, including decisions on the budget:
Code of Ethics: http://iap2usa.org/ethics
Core Values: http://iap2usa.org/corevalues
Community engagement may happen during various stages of the budget process and can provide citizens with various levels of impact over the decisions at hand.
A few typical scenarios include:
1) Make past and current budget information transparent
2) Help citizens learn about the budget process, incl. any opportunities to provide input
3) Solicit input to identify the public's preferences (this can take many forms, of course, from simple surveys to more complex deliberative public forums)
4) Let the public vote on certain aspects of the budget (e.g. to approve tax increases)
5) Give the public decision making power over a slice of the budget (a process known as participatory budgeting)
Pretty much all of these scenarios can be meaningfully supported, enhanced or extended through the use of (online) technology.
Which community engagement strategy is best for you depends on a number of factors, incl. how much room you have to accommodate public input, your overall goals, available resources etc.