The answer is very simple, but to explain why it is so, is a little tricky.
The best option is:
"What percent of the work is done?"
"What percent of the work is complete?"
The first sentence doesn't sound right. Even though percentages are numbers, they are not something you would include as a countable term. Percentages account to an amount in or for every hundred. Now, when you ask how far has the work progressed to far, we usually use percentages to reply. This is not something you can count. It is statistical, but uncountable. To use 'many', we need something to count, for example:
- How many units of work is over?
- How many days worth work is over?
- How many modules are complete?
Try this thread. This talks about why percentages are uncountable even though they are countable.