Your post seems designed to confuse us!
Are you referring to the MoT (which the main part of your post refers to), to the vehicle tax (which suddenly appears towards the end of your post) or to both?
You can renew the tax online if you've got the reference number number from the renewal letter. (Have you got it? Can someone go to your house to find the letter and email/text the number to you?). You can also renew the tax online with the number on your V5C [= 'log book']. (Did you take it with you? Can someone go to your house to find it?):
https://www.gov.uk/tax-disc
Staying with the tax for the moment, there are NO 'grace days' for LATE payment. However if you pay online (BEFORE the tax runs out) there's a 14-day grace period (from the expiry of the old disc) before you need to DISPLAY your new disc. In practice you could probably get away with longer since the police (and others) now largely rely upon a computer database, rather than on the actual display of a physical disc, for checking that a vehicle is taxed. (So much so that tax disks will cease to be used from 1st October anyway).
Further, failure to display a tax disc is a separate (and lesser) offence that failing to pay for one, so it wouldn't be a 'big issue' if you were unlucky enough to get caught. (However getting someone to forward the new disc from your home, to your address in Spain, would still be sensible if time allows).
The MoT rules, however, are far stricter. You can't drive a vehicle without a valid MoT certificate (where one is required by law) on UK roads unless it's being driven directly to an MoT test centre, so the advice to arrange a test close to your port of arrival seems good to me.
Of course the two sets of rules overlap in that you can't tax your car without a valid MoT. However you can tax your car right up until the very last day of the current MoT's validity (and the subsequent expiry of the MoT doesn't then invalidate the tax) so you've still got ten days to sort the tax out.