Chippy in Gmail
Select the part of a mail you want to hear and tap the right Option key; Chippy copies the selection the way Command C does, so it works in any browser.
Chippy has nothing special for Gmail, and it does not need to. It does the same thing in every app: when you tap the right Option key, it asks the app in front for a copy of whatever you have selected, exactly as Command C would, and reads that. Anywhere Command C copies your selection, Chippy reads it. Gmail in Safari, Chrome or any other browser is one of those places.
Reading a mail
Open the mail. Drag over the part you want to hear, a paragraph or the whole thing, and tap the right Option key. The panel appears with the text in it and starts reading, the spoken word lit as it goes. If the panel is already on screen, it stays where it is and reads there.
Select the next mail and tap again: the new text replaces the old one. Tap again on the same selection and it replays from the start, and nothing new is made.
Links and addresses are read as they are written. Chippy does not rewrite what you selected unless you turn on Make all Chippy selected text TTS friendly in Settings, which is off by default, so a long address is spoken as a long address. Select around it if you would rather not hear it.
The clipboard
One thing follows from how the copy works. Because Chippy asks for the selection with Command C, what you selected is on your clipboard afterwards, and what was there before is not. If you had copied something to paste later, copy it again after the read.
The same mechanism gives you a second way in. Copy some text with Command C, then tap the right Option key with nothing selected: Chippy reads what you copied, and the panel's header says from clipboard so you know which it was. Tap with nothing selected and nothing new on the clipboard, and the panel says there is nothing to speak, with no sound.