What do you think of this correlogram? do you think this weekly data is stationary? Unit-root test indicates rejects null of non-stationary (rejects null of unit root).

Thanks for your input.

I just wanted to follow up that the ADF test does not reject the null when I include two lags. That is important to remember when doing such analyses.
 A: The correlogram is descriptive .. the actual data is MORE inferential as there may be level shifts , local time trends , changes in parameters or error variance over time. Some of these conditions can induce a correlogram like the one you have WITHOUT the need for a regular difference or an AR(1) . Post a csv file of your original data and I will try and help further.
edited after receipt of data:
The OP communicated his data here http://autobox.com/dave/origdat . It contains 399 weekly values (recorded sporadically) for three series (Y being the dependent) recorded over 12 years.  The correlogram is only of value when you have data that is measured at fixed intervals of time ...clearly not the case here see http://www.itl.nist.gov/div898/handbook/eda/section3/eda35c.htm.  Apparently the OP didn't know this not-so minor detail . Only by closely examining the original data is one able to comment on the ill-constructed correlogram. There is nonsense and there is nonsense but the most non-sensical nonsense of them all is statistical nonsense and the correlogram presented by the OP is "statistical nonsense" since observations were not taken at fixed intervals. 
Time series analysis requires observations to be taken at fixed intervals such as every week/day/hour etc. Since this is a seasonal market and data is VERY intermittent one needs to modify the data in order to use appropriate statistical procedures to form a useful predictive model.
Noting that there was a consistent pattern of observations being recorded for 17 weeks every year (weeks 27 through 43) partly shown here , I elected to select these 17 weeks for each of the 12 years ..thus 204 triad observations were introduced to AUTOBOX , a piece of software that I have helped to develop to deal with "thorny issues" like this. The objective is to form a transfer function (multiple regression on steroids !) http://www.math.cts.nthu.edu.tw/download.php?filename=569_fe0ff1a2.pdf&dir=publish&title=Ruey+S.+Tsay-Lec1 is useful (ignoring the corner method suggestion) when Intervention Detection is incorporated. http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html .
So with 17 observations every year (much like modelling the demand for seasonal beers) we set out to identify 1) which of the two causals are significant and at what lags ; 2) what the effect is of the week of the season(17) while taking into account the effect of any trends/level/shifts/anomalies that might be encountered/identified. 
Following are the three time plots for the variables (Y,I,V)   and  . The (automatic) analysis yielded an Actual/Fit and Forecast ..here  . The Forecast for the next year is here  . The Actuals/Cleansed graph is here .
The equation that was developed included the V series and level shift indicators along with both week of the year and seasonal pulses and one-time anomalies  
The plot of the residuals is here  with statistical properties here .
In summary due to the large amount of "missing weeks" I used a consistent block of data over time to attempt to form a model to identify the importance of the two suggested supporting series. 
As an aside the fact that there are deterministic level shifts in the sample (204) values that I used explains why the acf suggests non-stationarity. It is broadly well known that an acf that exhibits non-stationarity as this does can arise from a series with a level shift or a sequence of level shifts or deterministic trends and thus is not-necessarily a license to difference no matter what your textbook/software tells you. To "prove" this simply simulate a white noise series  with a level/step change and notice what happens when you incrementally apply a larger shift.
A review of the empirically identified pulses  shows 7 of the last 17 values (188-20) have been identified as pulses and perhaps should be investigated (exploratory data analysis). The cleansing/adjustment of these 7 points is a factor in the visually obvious lower forecasts for the next "year" . 
