Would a simple OLS regression work with this example? First and foremost I would greatly appreciate any help you can provide me with.  
I am writing my undergraduate thesis on the rise of populism in France.  
The relationship I am trying to better understand is as follows:  Can we explain the rise in popularity of the National Front (political party) between 2000 and 2014 through the rising number of immigrant populations immigrating to France?  
The data I have so far are the average popularity ratings per year for the political party, and the change in immigrants in France? 
 A: If you are trying to establish a causal relationship between higher popularity or vote shares of the Front National with immigration then the answer is no. The reason is that you cannot construct a credible counterfactual, i.e. a comparison situation from which you can infer what would have happened to the vote share of the Front National in the absence of higher immigration. At best you can find a correlation between the two variables.
What you are trying to do is very difficult in practice and requires good data and appropriate estimation techniques. Running OLS only will not be sufficient because of omitted variables that affect both immigration and the Front National's vote share. This will bias your results and you may draw wrong conclusions from your data (here that explains the problem of omitted variables). Other econometric/statistical problems include reverse causality: immigration may increase the support of this right wing party but an increase of the support of this party may reduce immigration - in this case immigration is again correlated with the error term, i.e. your estimates will be biased.
In the perfect setting you would have two parallel worlds in which one has a France from 2000-2014 with immigration and the other one would have a France from 2000-2014 without immigration, all else equal. Then you could run plain OLS to uncover the causal effect of immigration of the Front National's vote share - of course we only have one world, unfortunately.
As an alternative: try to find county level data with vote shares for the Front National and identify counties that have immigration and some that don't. You can then compare those counties (with and without immigration) that have similar trends in the vote share for the Front National. Then you can apply a technique called difference in differences (see for example here for an explanation) which has higher hopes to uncover the effect of immigration on the vote share.
I say it has a higher hope because this model does not come assumption-free. You must make sure that you have included other time-varying factors in your difference-in-differences (DiD) regression that affect the vote share like unemployment, the vote share of other parties, etc., that you have two or more counties that are actually comparable in their pre-immigration characteristics, and that no other policies happened at the same time. The problem is that immigration is not a one-time event but immigrants flow over time. Yet DiD would be a good starting point and definitely at a very high standard for an undergraduate thesis.
