Lavender Damping Off In Window

It seems like the lavender plants I've just received are experiencing damping off. Is this due to temperature, and should I have kept the plants heated or are they shipped ready to go outside? I kept them indoors (it's been VERY rainy here) in a southern exposure.

It is more likely that the plants are either too wet, have too little light, or got too strong light too suddenly after being in the mail and in the dark too long or they could be showing delayed damage from the dark trip to you. Delayed damage would show up within a week of receiving them, with the plants very quickly starting to show damage after arriving. but then gradually getting better looking as they get more air and light.
They can go outside as long as it does not freeze. Keep them under the eaves in rainy weather and start in a shady location and gradually move them into the brighter spots as they get used to it. Take at least a week to move from shade to sun.