The typical way of verifying someone's debt is to pull a credit report. Given the nature of your request, the pull would be "soft" in the sense that it would not harm their credit score.
The advantage of using a credit report for debt verification is that you're involving an independent third party. By pulling the report (or using a credit verification service to do so), you eliminate the chance that the consumer can fabricate data (i.e. someone may show you a false statement). You also get access to the entire debt load a consumer is carrying, which would include both private and Federal student loans. Because it sounds like you intend to operate a charity to give people money if they can prove they have a student loan, there may be a very high incentive for people to fake the proof, which makes using a credit report very attractive.
Credit bureaus also provide electronic interfaces to facilitate this, which may help given you have stated this will be done for a large number of people.
Pulling someone's credit report requires you to get their permission, and to handle sensitive data about their identity. However, third-party services can mitigate the risk by handling the consent and sensitive PII on your behalf, and even potentially filter the report itself to just the lenders or products you're interested in.
With respect to the specific case of validating Federal student loans, the National Student Loan Data System exists as a clearinghouse of information available to validate loan details. However, this system is intended to support consumers (borrowers), not vendors or charities. There may be restrictions on it's use for the purpose you are proposing.