Here's a figure illustrating the potential force of the Amazon effect, put together by Kevin L. Kliesen at the St. Louis Fed. As the captions above the blue line show, e-commerce was 2.8% of retail sales, but has now risen to 10.4%. The blue line itself shows the price level for those items purchased via e-commerce, using 2009 as a base year. For example, from 2000 to 2009 this price index rose from a little above 90 to 100, implying an inflation rate for these goods of about 1% per year. But since 2009, the price index for goods purchased via e-commerce has actually been declining by about 1% per year.
It's interesting to consider the possibility that the falling prices for e-commerce retail may not be a pure deflation of prices. It might also reflect cost savings delivered because buying through increasing automated warehouses is becoming more cost-efficient, compared with standard wholesale and retail product chains.
For those who want details on this price index, it's the is the price deflator for “Electronic Shopping/Mail-Order Houses” produced by the US Bureau of Economic Analysis. It's in Table 7U. Chain-Type Price Indexes for BEA Retail and Food Services Sales, available here.
Of course, a 1% annual price decline on 10% of retail sales cannot, by itself, explain why overall inflation for the entire economy has remained so low. But if you allow for the possibility that e-commerce prices can also place pressure on bricks-and-mortar retailers to limit their own price increases, the Amazon effect could be a meaningful part of an overall explanation.