both species are known to cause allergies. So I would search for companies producing allergy tests and ask them. Or you could ask pest control companies if they had some cases regarding the stated species (or to call you if there is such a case so you can get some specimens).
The mould mites Tyrophagus putrescentiae and T. entomophagus are common pests of stored products, especially those with a high protein and fat content (meat, cheese, nuts and seeds, dried eggs. It feeds on the fungi that grow on the foodstuffs, and can become a pest of mycology laboratorie. If you need to start a colony of Tyrophagus mites, You can use cheese or nuts with high content of humidity infested with Aspergillus, a very common group of conidial fungi in the Ascomycota. Tyrophagus feed on Aspergillus and reproduce very easy. Let open the containers with food in a dark room with high humidity ( 80% RH and 26°C) to get natural infestation by Tyrophagus in two weeks.
For pure cultures of these species, try one of the researchers undertaking work on stored products acarology, such as Jan Hubert and Thomas Erban's at the Crop Research Institute, Biologically Active Substances in Crop Protection, Drnovska 507/73, Prague 6-Ruzyne, CZ-16106, Czechia. Jan's ResearchGate profile is https://www.researchgate.net/profile/Jan_Hubert