Run a help desk — Two ways in, one table
Setup offers two doors and you can take both.
An email address of its own, which you forward your real support address to. Mail arriving there becomes a ticket, replies thread onto it rather than starting a second one, and attachments come with it.
A chat bubble — one line of script on your website. It is created switched off unless you tick it here, because nothing publishes on your behalf.
Both land in the same table, so an answer is an answer wherever the question came from, and the customer who emailed you last month and used the bubble today is one person with one history.