For example, the I.R.S. could seek to revoke the foundation’s tax-exempt status. The agency could even do so retroactively, which would stick the foundation with a corporate tax bill going back years.

Ms. Underwood’s petition noted that Mr. Trump signed the foundation’s tax returns, in which he stated, under penalties of perjury, “that the foundation did not engage in transactions with interested parties, and that the foundation did not carry out political activity.”

That is the sort of information that investigators would weigh in determining whether he should be charged with filing a false tax return, according to attorneys who have worked on criminal cases that were investigated by the I.R.S. and brought by the Justice Department. They said that Mr. Trump could be especially vulnerable because of his repeated involvement with the foundation, directing it to spend money on specific activities.