Inaccessible crawl space

There could be several factors that may be contributing to the situation. It sounds like an air quality problem. This could be caused by airborne mold spores, chemical contamination from bad paint or flooring adhesives (VOC's) etc. , CO from a malfunctioning heating or A/C unit or something from outside getting in. I have had to address this problem with some customers in the past. We contracted an air quality testing from a local environmental testing lab. In your case, I would first consult with the management of the property. Explain the situation, confirm your conversations in writing, (document, document, document) and see if they will contract this service. If they won't for some reason, then I would have it done privately. Depending on the results and plan for mitigation, these costs should be reimbursed by the landlord. At minimum, it would give just cause to break the lease. If the results are bad and the management won't work with you, share the results with your local code enforcement office as they may help you and require the management to fix the problem. Once the type of irritant is identified, then the cause will be much easier to find. Good Luck.

You said both contractors "think" you have termites. They should know and be able to point out evidence like Tester101 pointed out. Both prices seem high for just basically spraying, especially since you don't have a slab that has to be drilled through. Near the front door, they could have just came in under the door. If they are subterranean, their should be evidence of how they got in - typically mud trails coming up from dirt, up the piers and into the house. They can also look for any things that are conducive to attracting them, such as any boards (pressure treated or not) that are in contact with both the house and the dirt. Just because you have seen a few doesn't mean they are already in your walls. A less expensive spraying might take care of it. I would call another guy and see what he thinks.