This country as a Constitutional organized system of law under the umbrella of an America, did not exist, so the Pilgrims by those standards could not have been illegal immigrants. However societal organization did exist on the lands the Pilgrims came onto.
The land the Pilgrims settled on was inhabited by Indians who had developed, and well functioning societies.
The Pilgrims moved onto these inhabited lands without permission of those existing societies they found there with no intentions of accepting the languages, traditions, and customs of the people that were there already inhabiting the lands. In that sense the Pilgrims were illegal immigrants.
So if one accepst that Indians are human and entitled to life, traditions, religion, and customs to the exclusion of outsiders, and entitled as human beings to form societies with rules, the Pilgrims were illegal immigrants, but if the Indians were seen as heathens, inhuman, unworthy to live and breathe, not created by God, then the Pilgrims could not have been illegal.