As per sendgrid documentation, the way Inbound parser webhook responds to failures are that it will retry to post the email to the configured endpoint for 3 days and will drop the email if undelivered.
The Parse API will POST the parsed email to a URL that you specify. If a POST is unsuccessful, SendGrid automatically queues and retries any POSTs that respond with a 5XX status. This prevents data loss for customers who have misconfigured their website or POST URL.
Respond with a 2xx status to the POST request to stop the email from retrying.
To avoid returning an error your link must return a 2xx HTTP code when the email is received. This response lets our system know that your link has received the email. It is then removed from our send queue. If we do not get a valid 2xx HTTP response, our servers will believe they have failed to deliver your message. Messages that cannot be delivered after 3 days will be dropped.
I would like to know if before dropping the undelivered emails, will sendgrid notify the sender that their email could not be delivered?
SendGrid won't send a notification before the email is dropped. However, the message will be dropped only after receiving an error from your Webhook URL and retrying to post the message for three days.