"The economic and health [crisis | crises] can be tackled together.”

No matter what you write, there is room for ambiguity, but mainly in a lawyerly sense; people who wanted to understand either of those sentences would understand.

If you choose the plural form

The economic and health crises can be tackled together.

it's fine grammatically, but it could also mean that there were several economic crises and several health crises.

If you write

the economic and health crisis can be tackled together

it has the ambiguity you already mentioned of perhaps implying that it is one crisis which is both an economical and a health one at the same time. But from the context with "tackled together" it becomes clear that these must be two separate crises. So it's understandable if not stylistically so good.

Another possibility thought it sounds a bit mouthy:

The economic and the health crisis can be tackled together.