In a universal suffrage democracy, mass agitation isn’t actually quite so important as it sometimes seems. The way to get a law passed is to bribe politicians to pass the laws that you write, and then those politicians will use the bribe money to agitate the party faithful to keep them in office. Politicians challenge one another for a chance to be a channel for that bribery. They will also sometimes get the chance to serve on helpfully labeled committees and caucuses that tell bribers from different industries whom they should funnel money to, for convenience purposes.
Bribing officials must always happen for private advantage at public expense, because otherwise there would be no motive to do it — and the advantage must come at the expense of some group.
The only successful conservative pressure group in the United States is the National Rifle Association. It has consistently fought for a single issue, amassed support for that issue, and ignored attempts to dilute its message for decades. It has the support of the explicitly written 2nd Amendment, but even that was tenuous for a while.
Would an anti-immigration pressure group be able to achieve similar results?
It might be possible for an organization to raise money from a broad base of supporters for the singular purpose of pressuring the federal government to tighten enforcement of immigration laws, and to reduce total immigration.
The organization would need to have enough pull to counteract the efforts of industries who benefit from low-wage labor, politicians who benefit from impoverished clients, and ideologues who believe in a world without borders.
However, by focusing on a single issue it might be able to have a big enough punch. Industry is not unanimously in favor of unlimited immigration; the organization could unite those businesses who do not benefit from slave-wage labor. The political parties’ support bases are not unanimously in favor of unlimited immigration: Republicans depend on support from law and order suburbanites; Democrats need votes from underclass whites and blacks both. Ideologues are largely hopeless, but largely uninfluential.
It could work. It’s just a question of numbers.