If an employee has any health issues, is he eligible to get an onsite opportunity?

There should not be any problem unless the visa process has a medical and you failed to clear it.

If your manager knows about your health situation and if you are taking frequent sick leaves then you won't get onsite easily. If manager doesn't know keep it like that. Manager will see what benefits the project and account is getting by sending anyone onsite. There is a billing angle as well. How much TCS can charge to client
If you are sick and taking leaves onsite then it won't work for the project. So chances are low

Also, this is my personal experience if your health is not keeping well then choose only those countries where you can get a doctor and the treatment quickly. A bad medical system will ruin your entire life.

But question is why you want onsite if you are facing health issues.

Also some countries have medical test as travel requirement. Not sure if that can impact the travel plan.