Only thermoplastics are processed by injection moulding. The the mould is opened, the part must be strong enough to keep its form, meaning a temperature not too high. To avoid waiting, plastic is injected at this lukewarm temperature just by a tremendous pressure. This is how injection achieves huge production rates.
So don't even suggest a thermoplastics injection moulding company to use a thermosetting plastic. It's not the same job.
Many thermoplastics can be loaded with chopped carbon fibres. Common ones include polyamides (PA), thermoplastic polyesters (PETP, PBTP), polyoxomethylene and ethylene-methylene (POM), polycarbonates (PC), more expensive polyamide-imide (PAI) and polyether-ether-ketones (PEEK) for high temperatures, and maybe polyethylene (PE) and propylene (PP).
It's rather banal in fact, and your injection moulding contractor should tell you what they're used to process.
Source of information, if you read German: Kunststofftabellen, by Carlowitz (89€ at Amazon and others).
If you read French and access a big library (unaffordable otherwise), look at the "Techniques de l'ingénieur", volumes about "Plastiques".
Books in other languages about injection moulding should tell about charges, including chopped carbon fibres.
Wiki?