To answer your questions:
Can I use machine learning, like a neural network, to forecast the sales of this product over the next year?
Yes - you can used neural networks, or some other generic ML method, to forecast sales. NNets have mixed results for time series, using other methods such as SVM or XGboost is not very common.
The are also other methods which are designed specifically for time series, such as ARIMA and Exponential Smoothing.
If possible, I like to perform the forecasting in R.
There is a package in R, the forecast package, which has many of these methods, including ARIMA, Exponential Smoothing, and some Neural Network models, and it is very easy to use.
6 data points is not really enough to use any sophisticated forecasting approach to forecast 7~18 steps ahead, you're better off just using a naive forecast (use the last month of the data as your best guess), or maybe a naive seasonal forecast (use current January to forecast next January, current February to forecast next February, etc...) but even then you need at least 12 months of data, so in your case would have a gap between 7 and 12 and can only forecast from 13 to 18.
Since it is sales data, it is very likely seasonal - so basically, you don't have enough data to forecast anything other than a seasonal naive of month 13 ~ 18.