CITES - How it works
CITES is a treaty among governments, which are called "Parties", now numbering 164. It is concerned only with international trade and has no power regarding domestic matters. If a country wants to allow or ban hunting, then that is the affair of the government. CITES only comes into the picture if species, alive or in parts, are proposed for international trade.
Each country has a CITES Management Authority and a Scientific Authority with which the CITES secretariat communicates.
Proposals for changes in the Appendices I, II and III, which set out the rules for trade in various species, can only be put forward by the Parties (the governments). They are voted on by the government delegations at the Conference of the Parties CoP, held every two years in various countries. Proposals have to be submitted 150 days before the CoP. Proposals can be seen on the CITES website, which provides all the information you are likely to want on the convention and its works.
NGOs can be observers at the CoPs, and have opportunities to express their views - which may influence the votes, although they are not allowed to vote.
Influence by NGOs and individuals can only be brought on the government making the proposal, and other governments which will have to vote on it, not on the CITES Secretariat, which is the "civil service" of the Convention.
CITES is often misunderstood as setting the conservation status of species, but that function belongs to the IUCN Red list. CITES deals only with species in international trade.
More information on how CITES works PDF