Do any house churches have a budget
Most house churches do not have a budget.  At the least, they do not need a budget.  There is no need for a paid pastor, building maintenance, etc.  If anything, a church may decide to take up a collection for different needs as led by the Holy Spirit.
Is it a problem to have a budget or not have a budget?
The principles we see in the New Testament have to do with generous and cheerful giving.  There is not much about proper structure and budgeting.  Budgets tend to promote legalism, tithing, and a centralized mentality.  The important part is that each member is giving to God's kingdom and we find it best to have accountability to encourage giving.