Can Ukrainians or U.S. soldiers stationed in Europe vote for the Czech municipal elections? No, that's not true: Only citizens of the European Union have the right to vote in municipal elections in any EU member state they reside in, under the same conditions as nationals of that country, according to the Charter of Fundamental Rights Of The European Union. This was also confirmed to Lead Stories by a spokesperson of the Czech Ministry Of Interior Affairs.
The claim appeared in a video (archived here) which was published on TikTok on September 11, 2024. The video contains a screenshot of a paragraph of a law in Czech, translated by Lead Stories staff as: "A person who has attained the age of 18, is a foreign citizen and is registered as a permanent resident in a municipality or military district within the territorial area of the region, if an international treaty which the Czech Republic is bound to and which has been promulgated so provides, shall also have the right referred to in Section 12(2)."
The text overlay on the video in Czech reads, as translated by Lead Stories staff:
How many Ukrainians and American soldiers are going to vote for county representatives
This is what the post looked like on TikTok at the time of writing: 
(Source: TikTok screenshot taken on Tue Sep 17 10:30:48 2024 UTC)
Municipal elections (archived here) will be held in the Czech Republic on September 20 and 21, 2024. The right to vote and stand as a candidate in municipal elections for EU citizens is enshrined in Article 40 of the Charter of Fundamental Rights (archived here). The article reads: "Every citizen of the Union has the right to vote and to stand as a candidate at municipal elections in the Member State in which he or she resides under the same conditions as nationals of that State."
This right (archived here) does not apply to Ukrainian refugees living in the Czech Republic or members of non-EU armies stationed in the Czech Republic since it is limited only to EU citizens living outside their home country within the EU.
Citizens of Ukraine, except for those who hold either Czech or other EU citizenship, cannot vote in elections in the Czech Republic (including municipal elections), Hana Mala, a spokeswoman of the Ministry of Interior Affairs (archived here), wrote to Lead Stories in an e-mail on September 13, 2024. Czech citizens have the right to vote in elections held in the Czech Republic (subject to other conditions, such as having reached the age of majority). In addition, citizens of other EU Member States can vote in municipal and European Parliament elections, Mala added.
The above TikTok video uses a screenshot of paragraph 13 of the Act on Regions No. 129/2000 Coll. (archived here), which states that foreign citizens with a permanent residence also have voting rights just as Czech citizens do, specified in a previous paragraph of the Act, number 12 (archived here), which the text also refers to. It also adds that this applies only if stipulated by an international treaty to which the Czech Republic is bound. No treaty between Ukraine and the Czech Republic would allow voting rights to Ukrainians residing in the country (archived here).
The right of EU citizens to vote in municipal elections and other specifications about voting can be found on several official government websites. It is stated, for example, as translated by Lead Stories staff, that "a citizen of an EU member state" who has a permanent or temporary residence in the municipality, who has applied to be included on the permanent electoral roll and also meets the age requirement, may also vote. At the polling station, the voter must prove not only his identity (with an identity card or travel document) but also his residence permit or temporary residence certificate (archived here).
The possibility and requirements for citizens of other EU member states to take part in the Czech municipal elections if they wish to do so are also documented here (archived here).
