Secure chat function with WordPress?

Hello, I hope I managed to put this in the correct forum!

My organisation has been using a pretty basic wordpress site for a long while and it’s worked fine. Now, we have become interested in implementing a chat function, to allow people getting in touch with us that way. Thing is, security is of paramount importance here. We are a volunteer organisation that assists people who are often in need of anonymous advice and who also quite often end up coming to us in order to escape an untenable living situation. We have a long and well established routine doing this sort of work via, for example, telephone but the internet is a fairly new arena for us and one that we approach very cautiously. We have seen that it would be beneficial for us to be available via chat, but we are worried about people being tracked via their interactions with us in a tool like this. So therefore I wanted to ask the community here: First off, are there stable chat plugins you’d recommend in general? Secondly, given our security concerns, what are your thoughts about having this sort of tool?

Please understand that we’re not all that internet tech savvy, so please go easy on us if the above questions come off as silly!