Yes, the government should be able to use open-source software, because the taxpayers pay for the websites to be produced in the first place. The government should not have a copyright on the things they do, because their work belongs to the people. Also, open-source software allows the public to take the government's work and modify it and improve on it.
Since open source software is highly customizable, governments need to take advantage of such opportunities. Plus, this software is more cost-effective than building a new platform or program from scratch. A government can save money and spend less time developing new platforms to help its citizens, which is beneficial in so many ways compared to building new programs.