Not all mites are harmful.
There are soil mites that don't cause harm to our plants.
But I can appreciate your concern if those are indeed spider mites, as they can do a tremendous amount of damage in a short amount of time.
I've used Neem oil on my more mature cacti, but I won't go as far as to advise if it is safe for seedlings.
One thing you might consider though, is bagging up you seedlings into a large ziplock bag or some other type of sealable clear enclosure and putting a piece of a used pet flea collar in the sealed bag with your plants.
You can even purchase a flea collar and just cut off a piece of it.
Try to ensure that the flea collar piece stays dry and don't put it on the soil surface that your plants are in.
The fumes from the collar will fill the inside of the sealed bag and should be sufficient to take care of any of your unwanted pests.