Can a person without any of the five physical senses attain arhatship?

A person without the five physical senses is still having the Vijnanas (識) of the five senses. A human is attributed with the five senses, 吾 is a Chinese character of I/me/myself, it's formed with upper a 5 (五), lower a mouth/opening (口). The formation of the five senses are with 3 factors: sense-Consciousness (識), sense-Organ (根), sense-Object (塵). A human born blind, deaf, dumb... whether born with defect or later damages, the five sense-Consciousnesses are always intact. What damaged are the sense-Organs, not the Vijnanas. Vijnanas cannot be destroyed, not even death. Hence a deceased person is still able to see, hear, smell... even with more acute and powerful capacity. Apart from the five senses, there is another additional, the mind/thinking organ, the 6th sense, called 意根. Therefore,

Can a person without any of the five physical senses attain arhatship?

the answer is yes.

I think it's been said that beings in the formless realms cannot attain enlightenment because they cannot perceive (hear, read, or otherwise learn) the Buddha's Dharma ... perhaps too they also can't perceive the three characteristics of existence.

This says,

The inhabitants of these realms are possessed entirely of mind. Having no physical body, they are unable to hear Dhamma teachings.

See also footnote 8 on this page,

An Abhidhamma text, the Vibhanga — drawing on a passage in DN 1 — mentions another level of devas of form not listed here: the Non-percipient beings, a level in which there is no perception at all. The commentary explains that they are not listed here because this attainment is open to hermits and seers outside the Buddha's teaching, and does not require virtue. This, however, ignores the fact that the four jhanas and four formless states are also open to people outside the Buddha's teaching; and although virtue helps with the attainment of jhana, it is nowhere stated in the Canon that the jhanas cannot be attained by a person whose virtue is not pure. A more likely reason for omitting the Non-percipient beings here is that this state, unlike the jhanas and four formless states, is not conducive to the arising of insight, and so would not be considered a desirable goal.

I don't know how to reconcile the above two statements, i.e. whether formless is the same as non-percipient.

People who are blind or deaf, or who are blind and deaf can learn to communicate. So I suppose this is only applicable to non-percipient beings in the formless realms: not people.

Cakkhupala Thera was able to hear the dharma.

“Avuso, how many conditions are there from which right view arises?”

“Avuso, right view arises from 2 conditions, that is: the voice of another and wise attention.

These, avuso, are the 2 conditions from which right view arises.”

Mahā Vedalla Sutta

So to develop right view in the Noble 8 Fold Path you need to be in a realm where you can learn the Dhamma.

If you achieve some stage of sainthood before being born in a formless realm though you do not have senses, you can progress to full enlightenment. I.e., when you have the 1st glimpse of Nirvana the path factors are established to progress unaided or further guidance, even in realms where you cannot find guidance.

You empirical understanding should be that validating the teaching. Only a Full or Private Buddha can understand the experiential result of meditation without guidance. Any disciple will need some guidance to interpret the experience hence this process is: Pariyatti (theory), Patipatti (practice), Pativedha (empirical validation). It is the theory you learn that is getting validated at least at intermediate stages. This is why you need to learn it from another (kalyāṇa-mittatā), i.e., learn from a Buddha or a Disciple. You need your sense faculties to learn.

In case of Chakkupala, blindness was not an impediment to learn, practice and empirically validate. In the story, the Ven. Sir practiced hard, with proper guidance and finally realised the teaching. Also one essential aspect of the practice is wise attention would have been present.