An arduino and thermistor sound good to me. I assume you need modulated temperature control rather than bang-bang (full-on, full-off) temprature control.
Acryllic has a max operating temperature of 80C so you can't make it from acryllic.
Why would you put the electronics inside the box? It's already small enough and you have two enormous lightbulbs which will fill most of it already, and possibly a fan. Not to mention it'll also be hot which isn't good for the electronics. The only thing you need inside the box are the heating elements and sensors.
You may not even be able to find a fan small enough that can also withstand those high temperatures. But I would think that at such close proximity to the light bulb (it's could be a LOT hotter than 90C in operation especially if you bang-bang instead of modulate it) the temperature gradient might be too intense if the bulbs are real close to the sample so you really would need a fan in that case.
Typically, an NTC thermistor is used for temperature sensing because technical reasons (material characteristics I think). Not a PTC. Thermistors are probably your best choice since they have the highest sensitivity but are non-linear. But you have a narrow, known operating temperature range so you can linearize and optimize for that range. RTDs are much more linear, repeatable, and accurate, but not as sensitive and so take additional signal conditioning to work. You probably don't need thermocouples either since you aren't running so hot that they are required and they take some additional signal conditioning as well.
Or you could use thermopiles to have contactless temperature sensing of the surface of interest. It's up to you. I don't know the caveats of your bioculture stuff but I assume you'll be more interested in surface temperature than air temperature and I don't know if you want the thermistor to be in contact with the same surface the culture will be on. I'd use surface heater than lighbulbs to be honest but maybe that's not suitable for cultures. Seems like it would be more effective though.
Note that common PVC wiring is not suitable for 90C operation.
In theory, you can reach any temperature inside any enclosed space if it is insulated well enough since you just keep adding energy to the system and it has nowhere to go (or can't get out fast enough). Maybe first try measuring how hot a 20W light bulb is in normal operation.