It depends. The term 'significance' is important here. Correlations can be done with any amount of data. The correlation between variables is then a fundamental property of the underlying data, regardless of how many data points you have.

However, as with all statistics, you need to be careful in how you phrase your ask. Do you plan on making a decision based on these data? If so, does the decision rely on rejecting the null hypothesis? If the answer to these questions is yes, then you need to understand how you then want to make a decision based on your data. For instance, there are typically two methods for making a statistical decision. This Penn online course link should give you a good understanding of some basic methods. 

You should also investigate the statistical power of the test to ascertain the quality. The second link goes over some power definition examples and should give you a good overview. Near the bottom third, you should look into how to calculate a sample size necessary to achieve a hypothesis test with a certain power. This should set you on your way into the wonderfully confusing, but super useful, world of statistics.

Feel free to DM me for direct questions.   

https://onlinecourses.science.psu.edu/stat500/node/44

https://onlinecourses.science.psu.edu/stat414/book/export/html/245

Sample is too short and results will not be generalizable. Simulate it then you will get some significance of estimates. If catagorical data then try for rank correlations only. You should try for partial coefficiens too.

Sample would not provide a valid result. It would be better to have some more samples.

I think this is interesting research.

Based on your data, you should be thinking in terms of "tests of association" and not necessarily "correlation", but the purpose is similar.

You might have some success with your sample size, but it will depend upon how they break down into groups on the various questions.  If you have only a couple of groups and the observations split evenly into the groups, you might find something interesting.

The first question you will have to struggle with is how to treat your "independent variable" scales (such as Innovation).  I assume with these that you will be summing or averaging responses for the several questions that make up the Innovation scale.  In this case, you will need to decide if Innovation is then treated as an ordinal variable or as a continuous variable.  If you end up several different unique responses (maybe more than 10 ???), you might be justified in treating the scale as continuous.  Otherwise, you should treat the scale as an ordinal variable.

Also remember the power of descriptive statistics to present your data:  median, box plot, 5-number summary, confidence interval (method appropriate for medians)

Some tests of association to consider.  Please investigate the assumptions and limitations of each before using.

No distinction between independent and dependent variable

Nominal - Nominal:  Chi-square test of association; Fisher exact test; G-test of association

Nominal - Ordinal:  Cochran-Armitage

Ordinal - Ordinal:  Linear-by-linear; Kendall correlation

Continuous - Continuous: Spearman correlation

Assumes ordinal or continuous variable is the dependent variable

Nominal - Continuous:  Kruskal-Wallis

Assumes nominal variable is the dependent variable

Logistic regression or multinomial logistic regression

Thank you Salvatore