The answer to your question is that, unless you received an actual W-2 from the National Institutes of Health, you won't want to enter this income under the regular W-2 TurboTax interview (which it seems like you are attempting to do, given that you ask about an "employer name" and EIN "number".)
Instead, you will enter this income on your tax return somewhat differently. While it is certainly still taxable income, a grant like this is not wage income, nor is it subject to Social Security or Medicare taxes.
For more details and a discussion, please refer to the following AnswerXchange posts:
https://ttlc.intuit.com/questions/3726385
https://ttlc.intuit.com/questions/3840431
The information found there will also show you how to (correctly) enter your NIH T32 grant income into TurboTax, which will then ultimately appear on your personal tax return.
Since my T32 went into effect, I noticed that my pay is not being taxed at all. Does this mean I will have to pay a large lump sum back in taxes at the end of the year?
Yes, you will owe Federal and potentially State taxes. In fact, you actually owe those taxes right now, but because they're not withheld from your paycheck, you're responsible for paying them on your own either monthly or quarterly; if you don't you could owe a penalty when you file.
See here for paying estimated taxes:
https://www.irs.gov/businesses/small-businesses-self-employed/estimated-taxes