Yes, the plan is required to keep documentation showing the dollar amount requested is an outstanding obligation for the participant. I believe what you are referring to with the immediate and heavy need is the safe harbor hardship reasons, which should be defined in the plan document. If you follow the predefined reasons (medical, purchase of a principal residence, tuition, etc.) then the reason is automatically determined to be valid but you must still provide proof of the amount being requested.
The participant is allowed to self certify that they do not have additional funds outside of the plan (IE, you don't need to get copies of their bank statements, denied loans, credit card statements, etc.) and you can accept their word that they have done their due diligence outside of the plan. However you still need proof that the amount being requested inside the plan is valid and will need documentation. See Treasury Regulation Sections 1.401(k)-1(d)(3)(iv)© and (D).
Treasury Regulation 1.401(k)-1(d)(3)(iii)(B)(1) indicates medical care that would be deductible under section 213(d) is Deemed to be an immediate and heavy financial need. If you cross reference this section to Regulation 1.213-1(1), "...a deduction is allowable only to individuals and only with respect to medical expenses actually paid during the taxable year, regardless of when the incident or event which occasioned the expenses occurred and regardless of the method of accounting employed by the taxpayer in making his income tax return. Thus, if the medical expenses are incurred but not paid during the taxable year, no deduction for such expenses shall be allowed for such year."
This leads me to believe if the amounts have not been paid by the participant than they can still be taken from the plan. However perhaps the plan may want to get an updated invoice to show the current balance due rather than relying on an outdated one?