Donkeys should be innoculated against equine influenza and tetanus. Check with your vet about how often as the guidance semes to vary. As for dog and cat innoculations, there is usualy an introductory stage with a few shots given over a short period then annual or bi-annual boosters.
Donkeys can suffer form the same sort of worms as horses and need regular worming. Until recently it was accepted practice to worm every 6 weeks and follow a programme of using wormers with different active ingredients (to reduce build up of tolerance by the worms) and also to use different wormers effective against different parasites at different times of the year. There are now longer lasting wormers which need only be used 4 times a year. There is also an increasing body of thought which says don't worm unless you have to - and adovcates having a worm count done on the dung and from those results determnining exactly what, if anything, you need to tackle.
In all cases I recommend you speak to either your vet or (perhaps more usual) whoever supplies wormer (quite often a feed merchant). Anyone who sells a wormer in the UK must be licensed and should therefore be able to give you advice.
Wormers can be bought as paste or granules, and sometimes as a liquid. Most donkeys are very good at spotting something strange in their food so may not take granules by this method. A paste comes in a 'delivery tube' that you pop in their mouth (via the gap between the front and back teeth) and squirt it in - they then have little choice but to swallow. An average 10 -11hh donkey will probably need approximately half a tube (depending on weight etc). Dosage guides will be on the packet.
Wormers are not only used to kill actual worms but also bot fly larvae (another possible donkey parasite)
Donkeys and lungworm
Many unknowing horse owners are anti-donkey as they beleive the donkey will give their horse / pony lungworm. If the donkey is wormed against lungworm this isn't going to happen (and I usally counter their comments by saying that my donkey is fully wormed but is their horse going to give my donkey anything?). There is some historical basis for their prejudice as, in the past, many donkeys were owned by travellers or 'little old ladies who knew no better' who did not bother to worm their donkey at all. Donkeys can carry lungworm without showing any symptoms, so if a traveller arrived with his donkey, and parked up near the 'big house's hunter' it could easily pass on the parasite.