Can we manage costs against an approved budget with multiple budget lines?

That definitely sound like something that can be set up with QuickBase.  You just need to think though the relationships you need in terms of "One to Many" and maybe draw them out on paper to get started.

For example
One does one budget line have multiple actual costs?

You used the word "Approved Budget"
Does One Approved Budget have many "Budget Lines"?

That is the starting point ... to think through the structure of your app.  ... which are the One's and which are the Many's for your tables.

Most people budget by the month or year, so one of the main tables you will want is a months table and a years table.  Then you will have budget lines, and actual expense lines tables, which are both children of the months / years.

Then with the summaries, you can compare where you are on your budgets.