The literal answer to your question, as put, is to use something like Teamviewer or pcAnywhere to access the computer at home from another computer.

But for short term use, I'd switch off the home pc and use webmail to get at my email accounts via a web browser.

Will you not be taking a portable computer with you? If so, why not set up your accounts in its email client? Or on your phone or tablet?

When I search the web for Mozilla Thunderbird, all I can find are sites to download M-T on the remote computer, which I do not want to do. Is it possible to directly log on to M-T on the web, to read and send emails?

rec-ebc said

When I search the web for Mozilla Thunderbird, all I can find are sites to download M-T on the remote computer, which I do not want to do. Is it possible to directly log on to M-T on the web, to read and send emails?

No it is not.

Thunderbird is an email client. A piece of software you run on a personal computer to work with email, just as a word processor is a piece of software for working with documents. Thunderbird stores your account settings and messages on the computer it is installed onto.

Thunderbird is not an email service or email provider. It has no visibility on the Web.

If you're using your own computer when travelling then you can put Thunderbird on it and use that to do email when away.

If you're using other people's computers then you don't want to leave your messages or your settings on these computers, so use webmail to access your provider's webmail sites. That is, use the browser to go to their websites and access your email there.