Where does the Bible support the idea of church budgets, for those traditions that think it does? Most churches work under a budget system, and have meetings every year to discuss this. What are their Biblical justifications for doing so?
Since of the Bible doesn't have a direct reference/listing of budgets, I believe the idea would come from teaching throughout.

Everything from Joseph's wisdom


  Genesis 41:35-37
  They should collect all the food of these good years that are coming and store up the grain under the authority of Pharaoh, to be kept in the cities for food. This food should be held in reserve for the country, to be used during the seven years of famine that will come upon Egypt, so that the country may not be ruined by the famine.”
  
  The plan seemed good to Pharaoh and to all his officials.


To the Temple tax


  Exodus 30:13-15
  Each one who crosses over to those already counted is to give a half shekel, according to the sanctuary shekel, which weighs twenty gerahs. This half shekel is an offering to the Lord. All who cross over, those twenty years old or more, are to give an offering to the Lord. The rich are not to give more than a half shekel and the poor are not to give less when you make the offering to the Lord to atone for your lives.


The tithe


  Exodus 23:19
  “Bring the best of the firstfruits of your soil to the house of the Lord your God...


The teaching of Jesus


  Luke 16:10-11
  “Whoever can be trusted with very little can also be trusted with much, and whoever is dishonest with very little will also be dishonest with much. So if you have not been trustworthy in handling worldly wealth, who will trust you with true riches?
  
  Luke 14:28
  “Suppose one of you wants to build a tower. Will he not first sit down and estimate the cost to see if he has enough money to complete it?
  
  Luke 14:31
  “Or suppose a king is about to go to war against another king. Will he not first sit down and consider whether he is able with ten thousand men to oppose the one coming against him with twenty thousand?


And Paul 


  1 Corinthians 4:2
  Moreover it is required in stewards, that a man be found faithful.


And many more.

So, I would submit that managing what we have been given wisely, which is what a budget is, is very biblical.