Not and get meaningful results. I refer you David Booth to any mathematical statistics book. David Booth

No, it would lead to wrong derivations as it involves selecting units from a population using a subjective (i.e., non-random) method. While inferential Statistics also involves discovering some property or general pattern about a large group by studying a smaller group of people in the hopes that the results will generalize to the larger group. Hence this will only lead to coverage of a very small population leaving along a very large population. And the results gained would absolutely different from what the population entails.

Check this recent paper:

Article Inference Using Non-Random Samples? Stop Right There!

For the second half of the 20th century inference from a sample generally depended on the probability-of-selection-based approach.  However, William Cochran, in 1977, noted that Ken Brewer, in 1963, and Richard Royall, in 1970, introduced the use of the prediction-based approach, with cutoff sampling.  Other methods developed after that.  Now we see people using pseudo-random sampling and prediction approaches with multiple covariates to make inferences from nonprobability samples.  In comments to such a paper I note that prediction using one predictor, the same data item in a previous census, can be very effective: 

https://www.researchgate.net/publication/316867475_Inference_for_Nonprobability_Samples/comments

If you use the methods of that paper, you are likely to be in for a lot of work, and may not find good data for suitable covariates. In the case of repeated sample and census surveys on which I commented, results are very good, but that is not the usual situation either.

Note that in https://www.researchgate.net/publication/316867475_Inference_for_Nonprobability_Samples/comments they show this link:

DOI: 10.1214/16-STS598

Which moved here means: 

http://dx.doi.org/10.1214/16-STS598

That will take you to the paper. (Richard Valliant has at least one other such paper on ResearchGate on this. J. Michael Brick has other nonprobability sample work on ResearchGate. And I suppose there are other such authors on ResearchGate and/or elsewhere. Note that Richard Valliant is giving the Morris Hansen Lecture this year.)

I have had a lot of success at the US Energy Information Administration using repeated sample surveys with occasional census surveys for predictor data, which should be useful for other such cases: "Application of Efficient Sampling with Prediction for Skewed Data," JSM 2022: 

https://www.researchgate.net/publication/362370770_Application_of_Efficient_Sampling_with_Prediction_for_Skewed_Data 

Work at the EIA used quasi-cutoff sampling for multiple attributes. When a ratio model is appropriate, as in cases I did at the EIA, though I also used some multiple regression there, we see fewer problems with obtaining good predicted-yi values. There is no collinearity or other such problems, and heteroscedasticity in regression is as it should be. See https://www.researchgate.net/publication/354854317_WHEN_WOULD_HETEROSCEDASTICITY_IN_REGRESSION_OCCUR.

Cheers - Jim

Another reference:

Valliant, R, Dever, JA, and Kreuter, F(2018), Practical Tools for Designing and Weighting Survey Samples, 2nd ed, Springer, Chapter 18, “Nonprobability Sampling,” Section 18.4, “Approaches to Inference.”

However, for both the quasi-randomization and superpopulation modeling techniques (and hybrid technique) noted there, you may need data for covariates that you might not be able to obtain.