If the infestation is more than just a small spot - meaning the invasion has barely begun, so a very localized spot treatment or baiting can be used effectively, then there is no real alternative. To kill the termites effectively, poison sprays will have to be used - which whether tentless or tented (which in my opinion is the only effective way to kill an established colony), means moving them out of the house for at least 3 days - and in their condition, maybe more like a week to be safer. Discuss it with the fumigator AND their doctor - essentially all treatments for general infestations are potentially quite harmful to people with weakened systems, and especially those with respiratory problems or allergies.
Sounds like me like a good time for a week long cruise, second honeymoon, visit to someplace they have always wanted to visit, visit to a relative for a week, or a trial week at a retirement/assisted living facility you might have been considering looking forward to when they might not be able to live alone (if a room is available). You can pay the fumigator a bit additional to open the house up and air it out after the quarantine period, so most fumes should be gone by the time they get back - or pay a neighbor to open up daytimes and secure the house nights while they are gone.
Depending on where the termites are, and financial considerations, I have seen several jobs where chenical treatment was out of the question, either due to severe allergies and asthma in a resident, or due to ecological concern about using poisons, so the colony was dug out by brute force - frozen in place with nitrogen gas, then the siding stripped away to expose all the nest and trails, and all damaged wood removed and replaced, siding replaced. These were cases where the infestation could be tracked with sensitive acoustic monitors to specificd locations in the house, with only one or two places needing exposing to get all of them. Then of course, in-ground barrier treatment was applied all around the house, and reinspection every month for several months to be sure there were not further termites in the house that had been missed. Total cost was in the $5-8,000 range - about 10 times tenting, but worked for those homeowners.