Does the t-test require randomization? Can I use t test for a non-equivalent quasi-experimental design? As there is no randomization, can it violate the assumptions of the t-test?
What statistical technique should I use?
 A: I'm aware of two bases on which statistical inferences (including hypothesis tests) are typically constructed:
One is based on some form of random sampling of a population of interest (not necessarily simple random sampling, though that's the most common situation people derive tests for).
The other is based on randomization arguments (typically randomization to treatment).
In order to perform a typical hypothesis test, you need some basis on which to identify the distribution of some statistic under the null hypothesis - you need some kind of probabilistic model to be able to do this, and both situations I mention above (with perhaps some additional assumptions) supply a basis on which to do so.
Without some such suitable setup/assumption somewhat similar to these, there's unlikely to be any basis on which to reasonably calculate a probability of falling into a critical region under the null hypothesis. 
It's not clear from your question whether any such basis would be possible for your circumstances (most likely not). 
Nevertheless it is not unusual to see people applying statistical tests (or point estimation, or interval estimation) where there's no clear basis for applying either a random sampling or a randomization type of argument (indeed, often it's clear that there could not be such a basis). Rarely is an attempt to argue for such a basis even made - it's usually ignored completely. Do the results obtained really mean much of anything? For many such cases, I'd say the answer is probably not.
A: Here are a couple of papers that suggest that hypothesis tests and p-values are not entirely appropriate without randomization. I've included excerpts from the abstracts of each one as well.

*

*Can p-values be meaningfully interpreted without random sampling?

Observational studies involve data sets whose size is usually a
matter of convenience with results that reflect a number of potentially
confounding factors. In this situation, statistical testing is not appropriate
and p-values may be misleading



*Use and Misuse of p-Values in Designed and Observational Studies: Guide for Researchers and Reviewers

Besides the inferential errors that abound in the interpretation
of p-values, the probabilistic pre-conditions (i.e. random sampling or
equivalent) for using them at all are not often met by observational studies
in the social sciences.

The first paper also suggests what to do instead, and includes a flowchart to help assess what to do:

other more modern statistical tools should be used instead, including graphic analysis, computer-intensive methods, regression trees, and other procedures broadly classified as bioinformatics, data mining, and exploratory data analysis.

