In real scientific research, it is quite rare to have data that came from true random sampling. The data are almost always convenience samples. This primarily affects what population you can generalize to. That said, even if they were a convenience sample, they did come from somewhere, you just need to be clear about where and the limitations that implies. If you really believe your data aren't representative of anything, then your study is not going to be worthwhile on any level, but that probably isn't true1. Thus, it is often reasonable to consider your samples as drawn from somewhere and to use these standard tests, at least in a hedged or qualified sense.
There is a different philosophy of testing, however, that argues we should move away from those assumptions and the tests that rely on them. Tukey was an advocate of this. Instead, most experimental research is considered (internally) valid because the study units (e.g., patients) were randomly assigned to the arms. Given this, you can use permutation tests, that mostly only assume the randomization was done correctly. The counterargument to worrying too much about this is that permutation tests will typically show the same thing as the corresponding classical tests, and are more work to perform. So again, standard tests may be acceptable.
1. For more along these lines, it may help to read my answer here: Identifying the population and samples in a study.