Nine rights groups, a former leader of the opposition in parliament and three prominent lawyers lodged the petition at the Constitutional Court on Thursday -- the second court challenge to the law.
The petitioners - which include Chapter Four, Uganda's most prominent rights group - say the law regulates online behavior in a "vague and ambiguous manner".
Chapter Four's acting Executive Director, Anthony Masake, told AFP that the new law's "strict and vague authorization standards" mean that journalists will never know when they are crossing a line by collecting information on people they are reporting on.
Legal experts have warned that the law will be used to target government critics who are already operating in a shrinking civic space.
The amendment to the Computer Misuse Act, signed into law by Ugandan President Yoweri Museveni last month, has been criticized by Amnesty International, which has called for the "draconian" legislation to be scrapped.
People convicted under the law are barred from holding public office for 10 years, which Amnesty warned was a way of reinforcing state control over online freedom of expression, including by political opposition groups.
Offenders also face fines of up to 15 million Ugandan shillings (about $3,900) and prison terms of up to seven years.
Uganda has seen a series of crackdowns on those opposed to Museveni's rule, particularly around the 2021 election, with journalists attacked, lawyers jailed, vote monitors prosecuted, the internet shut down and opposition leaders violently muzzled.