Called „wireless bridge“. Many travel router or home router offer an operating mode for this. You configure the SSID/passwd where the device should connect to. One or more Ethernet ports let you connect your device. (This layer 2, so the device should be able to get an IP from your existing LAN like any other device. Zyxel N300 is good and cheap.
A simple approach would be to just run an IMAP server. For receiving you use a script using POP3 to download newly arrived emails (which is run every 5 minutes for example). If your server is offline, nothing gets lost.
For the sending side you can either use your mail ISP’s SMTP server directly from the client or you install a local relay as a mitm to the ISP’s SMTP.