There are quite a few reputable groomers that will come to your door and groom your dog(s) for you if you don' t feel like travelling.
It is actually against the law for taxi drivers to refuse customers when the "hire light" is on and you can file a complaint with officials if necessary. As flubber has already mentioned there is a $5 per pet surcharge for that and please always remember to clear after your animals. Another options would be hiring a van, (try Easy-Van: 26640000 or Oriental: 23802211) which is a very popular way of transportation for people with large dogs, these van drivers are usually more pet-friendly than most cab drivers. A ride from Wan Chai to anywhere in mid-levels cost around 50.
As for recommendation, try The Ark in Sai Ying Pun, DogOneLife in Causeway Bay and a small place called Pet House in Happy Valley.