Yes, you can order research chemicals online. You can do that right here, in fact. It is perfectly legal to buy non-regulated chemicals, lab supplies and the equipment you need to advance your understanding of chemical science.
For practical reasons, most scientists need to buy their pharmacology supplies online, as they cannot buy them locally. You cannot buy all designer drugs online, though. Several are ‘controlled substances’ in many jurisdictions, and you should take care not to attempt to order or import any chemicals identified as stimulants, cannabinoids, sedatives, opioids or benzodiazepines in the place where you live or work.