If the company is simply renting “unfurnished” residential accommodation, then furnishing it and renting it on as “furnished” residential accommodation, their supply to the Local Authority (LA) will be exempt from VAT – in the same way as the landlord’s supply to them would be exempt from VAT. Income which is exempt from VAT does not count towards the £85k threshold.
However – are they providing anything else? You mention that the accommodation is “semi-independent” – so are there also (for example) any supplies of care, catering or cleaning – or maintenance other than that for which a landlord would normally be responsible? You mention that the charges are processed weekly – so are the supplies of accommodation potentially short-term, which might be more aligned to provision by B&Bs? Or is the company just a management agency – i.e. are there two separate leases (landlord to company and company to LA), or is there just one lease (landlord to LA) with the company just handling rents and providing furnishing? The exact nature of what is being supplied needs to be known in order to be certain of the VAT liability, and the key to this will probably be in the contract between the company and the LA. It may be that there is nothing extra being provided by the company – but if that was the case, I can’t see why the LA wouldn’t contract with the landlords directly?
In the event that the exact nature of the supplies make the charges potentially liable to VAT, bear in mind that LAs are generally able to recover from HMRC any VAT that they are charged. This means that (subject to any contractual terms) if VAT registered, the company could add VAT to their charges at no cost to the LA. This would then entitle the company to reclaim the VAT incurred on furniture and any other VATable purchases … so VAT registration might be beneficial. But this is only possible if the income is not exempt.