Is there a way to set up our web chat so that only certain people receive the chat message? Do you need to add to skill levels?
If your team is assigned to service the web chat, the agents in that team will receive the chat messages. If you do not want anyone to receive these messages, or for that matter, you want other agents to receive them, you will need to adjust the skills levels for the web chat.
Best practice is to set skills at intervals of 25 (e.g., 100, 75, 50). What this will do is cause the web chat scenario to find an agent with a skill level of 100 first, then 75, and then 50. The higher-skilled agents will get the chat messages before the lower-skilled agents, and the agents with no skill will not be served a chat.
For more information, see the Contact Center Administrator Guide, section Skill Levels.