The short answer is no but some seem to do so.
As described in another post, many academies have funding agrements which require then to consent to being named in a pupil’s statement of SEN (and thus, in effect, to admit that pupil) unless admitting them would be incompatible with the efficient education of other children and there are no reasonable steps that can be taken to remove that incompatibility.
None, as far as I know (do tell me if I’ve missed any) have funding agreements which would allow them to refuse admission in that way simply because some numerical threshold has been reached. In other words, they cannot just set a quota, say one pupil with a statement of SEN per class, or something like that.
And yet some try to impose such a limit. I have personally seen several which have tried to do that. In each case, they failed in the end, either because the case went to a SENDIST and the Tribunal entirely rejected the academy’s approach; or because when the academy asked the Secretary of State to adjudicate (under the funding agreement) on whether the academy should have to ‘consent’ the Secretary of State ruled against the academy.
In both types of situation, the person deciding the issue (tribunal or Secretary of State) found against the academy (clearly rightly in my view). But that did not stop the academies in question trying it on. And I have every reason to think that they (and others) continue with such practices other than when caught out and challenged.
The simple point is that, provided they have the general form of words in their funding agreement (as above) they cannot lawfully set numeric quotas. The issue has to be, for each particular applicant child, would admitting them cause prejudice to the efficient education of other children, and – if so – would their be reasonable steps which could be taken to remove that incompatibility.
Setting quotas (say of one child with a statement per class, which I have seen) is completely inconsistent with that. It fails to consider ‘incompatibility’ or ‘reasonable steps’ as required. It is, moreover highly likely to be unlawful disability discrimination (assuming that the pupils in question as disabled – as to which see another post). That is because it assumes that children with a statement are a detriment and it assumes they are all equally so. Plainly, neither is correct and the making of the assumption is clearly wrong.
So, in summary: if academies are setting quotas then (unless they have specific provision in their funding agreement allowing for it) that is likely to be a breach of the funding agreement. And, even if they have provisions in their funding agreements allowing for it, those provisions are quite likely to be unlawful under the Equality Act (and could be legally challenged on that basis).
I would be interested to know of any examples you come across.