Will plants accumulate toxins from soil contaminated by toxic substances like paint thinner/lead paint?
You could use other plants to extract harmful substances from the soil. There are plants which are called hyperaccumulators which absorb high amounts of contaminants from the soil. 
Have a look at at Phytoremediation where the process of phytoextraction is described in detail.

Normally phyotextraction needs several generations of plants to be grown in order to be effective. Consequently you could just plant your vegetables as usual but take samples and test them for heavy metals or other contaminants of concern.