>it create difficulty for undocumented people to access the web
Why is this such a sticking point in US politics? If the "undocumented" people aren't supposed to be in the country in the first place, why should rest of society cater to them? Even if you're against age verification for other reasons, dragging in the immigration angle is just going to alienate the other half of the population who don't share your view on undocumented people, and is a great way to turn a non-partisan issue into a partisan one. It's kind of like campaigning for medicare for all, and then listing "free abortions and gender affirming surgery" as one of the arguments for it.
> why should rest of society cater to them?
Because these undocumented people are still humans. They deserve access to information services. It's as simple as that.
I don't think it's "catering to them" to avoid passing laws that impose undue burden. For example, if you passed a law requiring a US passport to buy food in the US, and made it so all restaurants and grocery stores are required to check passports before selling food to anyone, I would be opposed to that law, and part of the reason is that I don't think it should be hard for anyone to get food, whether they have a US passport or not.
"Undocumented" doesn't mean "residing illegally" anyway, it just means "lacking documents", which is a state that many perfectly legitimate US citizens find themselves in. But we should want people who are here illegally and everyone else to be able to use the world wide web and computers regardless of their legal status, just like everyone should be allowed to eat and buy food regardless of their legal status, because that's just basic humanity.
What if they are supposed to be in the country, but they are undocumented?
This means "not having documents". It's not a synonym for "illegal immigrant".
There are many ways to not have a state or national ID document in the USA. You might simply not have a driver license or passport. That's totally legal. You might be in the country temporarily for business or as a tourist. The constitution applies to all of these people.