The part of the constitution that protects the rights of interest groups to gather and petition the government is the First constitutional amendment.
Below is the text, taken from the first amendment, which strengthens this answer.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
