PayPal integration: Add a service charge

I use my account for business and I noticed there is not a way to pass on a service fee to a customer. 

Do any of the other payment methods allow you to pass on a service fee to the customer?

While you're configuring your PayPal integration you are able to add tax or shipping cost to the total: 

How-to-Integrate-Form-with-Paypal-Standard-

Here are some guides that will also help you: 

https://www.jotform.com/help/304-How-to-Add-Shipping-Cost-to-Payment-Forms

https://www.jotform.com/help/188-How-to-add-Shipping-to-your-Paypal-standard-account

https://www.jotform.com/help/296-How-to-Add-Tax-to-Your-Payment-Form

However, this changes based on the payment type you have selected, if you're using the "custom amount" option you will not be able to set  up an additional cost, you will need to perform calculations and add the additional cost, these guides will help you doing so: 

https://www.jotform.com/help/259-How-to-Perform-Form-Calculation-Using-a-Widget

https://www.jotform.com/help/275-How-to-Pass-a-Calculation-to-a-Payment-Field 

I hope this helps. 

Does adding tax just add on an additional fee?
...

My worry is that it would come up as tax and I wouldn't be able to access
the money.
...

The tax works as a percentage of your total. You just supply the percentage. 

The shipping cost is a fixed amount.

For both you can modify the labels. So instead of saying "tax" or "shipping" you can customize it as "service charge" or "service fee".