Typically the Government doesn't hire outside expert witnesses for their purposes unless the specialty is atypically nuanced. As a prosecution expert witnesses are part of the jurisidiction's crime lab and they specifically handled the evidence in question. The generic term will depend based on the jurisdictions of the handling, but are usually departments of the policing authorities working the case, though in the United States, a more expert witness will usually come from a federal agency (usually the FBI, but depending on the case, other agencies may be better suited).
The defense or plaintiff (if either isn't the government) typically have access to private forensics labs that can examine the evidence for a fee (In the U.S., the defense always has a right to examine all evidence found in an investigation as well as all government results) and the expert who handled the evidence might be on retainer by the law firm, if they aren't big enough to have their own labs. Typically they will offer expert witnesses for a fee, usually billed to the law firm who is reimbursed by the client who's case needed it (typically, a lawyer and client will set up a trust fund that will hold a sum of money for the attorneys fees for immediate use, and may have it returned at case conclusion if there's still something in there or reimbursed in a victory (typically the victorious side is granted legal fees from the losing side in addition to compensatory and punitive damages... unless the victorious side is the government).
There is no term for these businesses as an expert witness is a witness who is qualified to give an opinion on a matter of a limited field of knowledge. So you may need a DNA foresnics specialist to test the DNA and testify to the findings, but if your case involves it as a matter of evidence, you may need to find something as odd as an expert on Beanie Babies to verify that the stolen plush is a rare defective character and worth a lot of money to collectors. They need not be certified by any official documented education or training, though it doesn't hurt. The process for qualifing an expert witness is called Voir Dire, which basically is a brief period where the side calling the witness can present the qualifications and the opposing council can object on any problems with the witness giving expert testimony. If they have no objections or all their objections are satisfied by the presenting council, the witness is allowed to give testimony.
For a good example of voir diring the witness, check out the film "My Cousin Vinny" which includes a scene where the titular Lawyer has to voir dire an out of work hairdresser as an expert witness in general automotive maintenance (It makes sense in context. Its used in legal schools as one of the best films about trial procedures ever made, and this scene is the crown jewel as to why.).