The "Computer Misuse (Amendment) Act", which came into force last week, bans people from using a computer to send any information that might ridicule or degrade someone.
It forbids the recording or videoing of anyone without their consent, among other clauses, and has penalties ranging from fines to jail time.
Ruling party officials and other supporters of the legislation have argued it will curb hate speech, protect children and stop the sharing of false or malicious information.
President Yoweri Museveni, who signed the bill into law on Thursday, has regularly complained about what he calls lies against his government on social media.
Rights groups have called the law draconian, and said it adds to the arsenal authorities use to target critical commentators and punish independent media.
In their filing to the Constitutional Court, the 13 petitioners said the law was unconstitutional, ambiguous and criminalised freedom of expression.
Peter Arinaitwe, a lawyer representing one of the petitioners, Alternative Digitalk Limited, said the law's vague language could punish legitimate communications.
"Citizens have a right to express themselves. It is inherent. It is a right given to us by nature," he said.
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