They clearly can and frequently do
The USA is peculiar compared to most democracies in how it handles voting and the rules related to voting.
This is partly because the Constitution and federal law leaves most detailed decisions about how elections are handled to the states and has very few restrictions on how they exercise those responsibilities. This creates many opportunities for "suppression".
One particular oddity that creates the opportunity for suppression compared to most democracies is the way voter registration is managed. Most countries have automatic registration or compulsory registration. In the USA registration is mostly voluntary and subject to local rules. This creates the possibility of suppression before any voting even happens. Explicit criteria preventing registration based on sex and race would be a violation of the 1965 Voter Rights act, but there is plenty of scope for measures that achieve that goal without explicit discrimination.
The USA is also unusual in allowing states to choose boundaries for voting districts however they want. How district boundaries are drawn can, in effect, disenfranchise certain groups (both Democrats and Republicans have done this a lot). While the Voting Rights Act prohibited redistricting for partisan or racial advantage, the Supreme Court recently nullified the requirement of some southern states to submit redistricting decisions to the Federal government before implementing them, reducing the restrictions.
But attempts by states to actively suppress the effect of votes by minorities or opponents continue. This should be obvious from the surprise recent Supreme Court ruling that nullified the 2020 Alabama redistricting based on an attempt to reduce the impact of black voters. Previous rulings set a very high standard for ruling that redistricting was unfair (leaving many commentators wondering just how extreme states had to be before the court would act). The case shows that states are still attempting to do it but that they have to behave egregiously to get stopped by existing laws. The key point is that remains a great deal of leeway in federal law that allows states to suppress votes via redistricting.
Also the way Electoral College votes in presidential elections are distributed is chosen by states (most states choose to allocate all the voters to the candidate with the local plurality but some allocate them proportionally). It isn't obvious that this counts a suppression of votes, but it clearly leads to many elections where the winner didn't win the popular vote leading to votes in smaller, rural states having more influence.
States can also choose the processes for voting in ways that make it harder for some groups to vote. Barriers to registration may discriminate; barriers to physical access to voting may be problematic as the accessible choice of voting method may be. The legal arguments around early and postal voting surrounding the 2020 presidential election illustrate that the rules are sufficiently variable to create the possibility of bias against certain groups. Some states have long had universal postal voting without issues. Others have fought against making postal voting easier (claiming it creates the possibility for more fraud but, in reality, because many thought that Democrats were more likely to vote postally). Other states have deliberately restricted the location of voting centres or early ballot drop off centres for arbitrary reasons that often seem to make it harder for people in opposition supporting areas to vote. There are no standards mandating equal access to voting facilities and states have plenty of leeway to exploit this for partisan advantage.
States also have wide freedoms to define identity requirements for voter registration or actual voting leaving them with a lot of scope to discriminate. Restrictions on registration would be almost unimaginable in most other countries as the process is automatic or compulsory but US States are free to set standards that can and do suppress votes from some minorities or groups (not everyone has an acceptable photo ID, for example, and the groups that don't are often Democrat leaning). While it is not uncommon for other countries to require IDs to vote, this is often accompanied by widespread availability of free photo ID on application before such restrictions are applied to minimise the possibility of bias against certain groups.
There are even cases when large groups can be prevented from voting. there are nearly 6 million voters with spent felony convictions. Some states bar them permanently from voting, some allow them to vote with various restrictions. Some attempt to suppress their votes despite having passed state laws in ballot initiatives to allow previous felons to vote (as Ron de Santis recently did in Florida).
Summary
The laxity of federal rules leaves a great deal of space for states to manipulate voting registration, voting access and voting boundaries. And many states choose to exercise that freedom to suppress votes and their impact from opposition supporters. That this happens is attested by the numerous legal challenges (though only a few are successful). All of this is extraordinarily unusual by the standards in other democracies.
