The first amendment is not related. I think you are talking about the fourth amendment (protection from unreasonable searches and seizures). In this case the online data is stored by the provider, and there is no private location for which to obtain a warrant. The provider's database is not your private domicile. Legally, it is no different from a cop asking a store for footage from their cameras.
Of course, the claim is that it should not be considered this way, because it is bad for privacy. But the reasoning that led here is pretty comprehensible.
Legally, it is probably the case that the laws are just not written to satisfy those of us who want privacy, right? The “ubiquitous privatized surveillance” industry came about after our government lost the institutional capability to pass new laws that help people (around the turn of the century).
4A says "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures" so I would think the provider of the service could consider their data to be "their papers/effects" but is the provider a member of "the people" if it's not a sole proprietor?
Yes, providers can absolutely deny requests that aren't lawful. A company is within its right to say the data is their property stored on company property and a warrant is required to search it.
They probably could refuse, but isn’t selling access to surveillance information about you part of their business model? As they say, “we value your privacy…”
Of course, the claim is that it should not be considered this way, because it is bad for privacy. But the reasoning that led here is pretty comprehensible.