Random sampling is a key element in most statistical methods. Yet, using a random sample can result in other complications. This can be just the nature of random: I have an experiment with two treatments and I pull plants randomly from two greenhouses. I am very unlucky, and all the plants for treatment A come from greenhouse 2, and all for treatment B come from greenhouse 1. Mostly what happens in this case is that the plants are re-randomized. Yet excluding one or more possible randomizations results in a non-random randomization. Another complication that arises in observational treatments (a treatment that is not applied: like sex) is that one or more of the states is uncommon. So I can get 400 surveys of white males between the ages of 20 and 30 that make 40-60K/year, but in this sample I have 2 female pacific islanders that make 200K or more per year. So I need to purposively select participants in order to get a good sample of the latter case without being swamped by the number of candidates in the former case.

Another thought: The purpose of randomization is to eliminate intentional and unintentional biases in the data. Even if you do not go to a random number generator to select a sample, is your sampling protocol such that most reasonable readers will not question a bias in your sample, or can you recognize such and discuss?

Finally, many studies replace random with arbitrary. I assign each plant a number and use a random number generator to select plants, versus I grab plants out of the greenhouse as I find them. Some studies use restrictive randomization. Block designs are an example.

Yes you can, while conducting any research its important to look at the feasibility of doing so, going everything by the book is not always possible in research.

The purpose of randomness is to attain a sample representative of the population under study. All inferential statistical tests and estimation procedures are based on probabilities that are derived from random sampling. Thus, when randomness does not exist, representative nature of the sample becomes an assumption. For example, your conclusion is correct assuming the sample is representative of the population. You can make that assumption more reasonable by establishing the representativeness of the sample. This can using one or both of two primary ways. The first is to compare your sample statistics to known parameters. For example, if you are doing a sample of people from a specific local, compare the demographic characteristics of your sample to known population information (% of people with certain characteristics). The second way is to use a very large sample proportional to the population. While randomness is an assumption of every inferential procedure, it is often ignored and/or violated.

good information