"Can someone actually say they won't accept email, nowadays?"
Unfortunately, yes, they can. At least, when it comes to service of legal documents.
Service via email is "opt-in" and cannot be done without prior written consent, even if the parties already communicate via email.
So, for example, if you are sending them your Particluars of Claim or Defence etc (sorry, I'm not sure whether you're the Claimant or Defendant here), or any other document that has to be "served" then they can insist that you do this either by personal delivery or first class post (second class doesn't count).
But also, if they try and serve you via email then that doesn't count either unless you have already said that you will accept service via email.
But "service" has a very particular meaning and only applies to certain types of communication between you and the other party.
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"Do I need Word to do letters properly"
Definitely not.
My DH is currently involved with a legal case and he has written everything on a free version of Word called LibreOffice. MS Word has got an awful lot of advanced features but you really don't need all of those just to write some documents.
It's totally free and you can save documents as standard MS Word documents.
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"...or would I need a laptop?"
Either a laptop or a desktop (but you need a separate monitor with a desktop). Generally speaking, an equivalent specced desktop is normally cheaper than a similar laptop.
But you don't need to buy a new one, there are plenty of refurbished PCs for sale that are very cheap and more than enough to write a few letters.