I'm not a lawyer, I'm an adviser, so always take proper legal advice as well. But, from what I know thus far from my work and from the government guidelines on this, training materials and conversations with solicitors and barristers over the last year....
www.equalityhumanrights.com/advice-and-guidance/new-equality-act-guidance/equality-act-guidance-downloads/ is the best place for getting the right info on what is a breach of the Act and what to do about it. In the first instance, I'd always say to talk to the organisation concerned to make sure they realise there is a breach of the law happening in their service. They have an obligation then in law to act swiftly to investigate, let the person know the results of that investigation, and take any action. And do so in a way that respects confidentiality, not necessarily on a public forum. People are free to take further action if they need to. The law is now quite powerful, and if a person feels humiliated or intimidated by aggressiveness or negative behaviour towards them (from staff, volunteers or service users) relating to a protected characteristic (whether disability, gender, sexuality, race or otherwise), that is normally seen as quite enough evidence that a breach has happened.
All service providers also have a duty now to make Reasonable Adjustments to ensure that all disabled people have an equal chance of contributing, and an equal chance of accessing the service and joining in with that service. Are there easy-read documents, or is there assistance available for interpreting them, fast? Are there the right adaptations for people with visual impairment if needs be, etc. The exception would be if it changed the nature of the service by doing this. Or if it cost so much to make the changes that it would near-bankrupt the service concerned (though this is a tougher one to prove these days, since if you're running a service, they're supposed to factor in disability needs at the start, not realise there's no cash for it half way through. Otherwise it's like saying "ooo, there's no cash for fire extinguishers or alarms in the building - never mind, we won't have any").
www.equalityhumanrights.com/uploaded_files/EqualityAct/service_users_businesses.pdf p 81+ gives details on how to take action if required. Alternative Dispute Resolution is one possibility. The other is to take it to court within six months of the incident(s) happening.
From the service provider's point of view, they would have to be seen by a court to have carried out advance checks that disabled people can indeed access things in a fair way, consulted with the right authorities, decided how to spend their money wisely, set up written and active policies for disability matters etc, put in the right training and information for all staff and volunteers, and actively monitor for reports of breaches so they can take fast action. All easy stuff to do, and help from people like me is always available, as they already know. 'Tis why I am asked to advise a large number of online services on this stuff - to aid them in getting it right. It's a free service, as well, since it's supported by a community fund.
Also, let's say that service user A calls service user B a r n b***d on an online forum. It doesn't even matter that service user B isn't learning-disabled or black - the assumption that they are, and the abuse directed at them as a result, is a clear breach of the Equality Act (if that forum's managers don't take the right action when they find out).
And if behaviour is because of disability, it is a breach of the Equality Act to target that person for it. Thus someone with a learning disability cannot be mocked for not understanding a debate. Someone with autism cannot be mobbed by a group for not understanding social rules well enough or getting elements of social language wrong, etc. Someone with profound CP cannot be thrown out of a restaurant or treated in a worse way for (say) dribbling or 'upsetting' the other customers by looking different. Respect for disability has to be shown by all involved, and a service has to be given as fast and as effectively to each disabled person as they would for others (and indeed to each person with other protected characteristics).
Hope it helps a bit.