Intercom will do that for you, and as an added bonus you can configure different options such as only asking for an email address outside of the business hours you define so you can get back to them once you're back online. You can also automate responses for common questions.
It can be slightly more expensive that some other chat apps, but depending on how big your business is you can apply for early stage business pricing which is $49 per month for the first year. It works with Shopify off the shelf as well.
Check it out see what you think.
I think you should have a look around Shopify Ping https://help.shopify.com/en/manual/promoting-marketing/ping. Shopify Ping is a free iOS app that you can use to manage messages from your customers. Visitors to your online store can send messages using messaging channels like Shopify Chat, Facebook Messenger, or Apple Business Chat, and you can use Shopify Ping to read and respond to your messages from one place.
Hope it helps
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AiTrillion.com has a chatbot that asks your customer to leave an email address if you are not available. It is all in one Shopify app. Plus you don't need to install multiple apps. You can check it out.
Forefront Support offers chat functionality, but also provides the actual support as a service!
Agents will solve your customers problems, sell them on products, and make them feel welcome at your store. You can pay per conversation or per response.
Try it out https://apps.shopify.com/forefront-support