Hi Lainey - This is something I worry about, too, on behalf of my dad who is mam's carer. If he was ill I'm afraid I wouldn't be able to look after my mam and would need to place her into emergency respite.
Things have changed a little for us now, but initially what I did was to email social services adult section and explain the situation. I was then given a telephone number of the duty social worker team (can't remember their proper title) so that I had a way of contacting them 24 hours a day if there was an emergency. They would then arrange things.
This might be useful for you to do.
However, I'm a little concerned at the level of care you and your sister are providing together, which is wonderful, but perhaps you could both do with some general help and respite from caring? Social services is still the place to go, to request both an assessment of your mum's needs, and an assessment of your and your sister's needs, as her carers, too. I'm fairly certain that you will all be entitled to this assessment.
I could say more but don't want to inflict information overload! Do keep posting, and others more knowlegeable than me will be along shortly with more advice and support, too.