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If you want to include more salespeople and want to be able to compare salespeople between them, you could plot greyed out lines of all salespeople in each cell (with the overlayed colored line of the salesperson corresponding to the row). This is far from being a perfect solution, though. Here's an example, with 12 salespeople:

Compare to the same graph without the grey lines (but with a background grid):

Passed a certain number of salespeople, I suspect that there would be just too much information to show, and that any kind of visualization would require to use some interactive visualization, or to generate multiple graphs, or to abandon the "salesperson" facet altogether and to regroup them by category (i.e. "salespeople from sector A, B, and C", or whatever classification of interest you may have).

If you want to include more salespeople and want to be able to compare salespeople between them, you could plot greyed out lines of all salespeople in each cell (with the overlaid colored line of the salesperson corresponding to the row). This is far from being a perfect solution, though. Here's an example, with 12 salespeople:

Compare with the same graph without the grey lines (but with a background grid):

Past a certain number of salespeople, I suspect that there would be just too much information to show, and that any kind of visualization would require to use some interactive visualization, or to generate multiple graphs, or to abandon the "salesperson" facet altogether and to regroup them by category (i.e. "salespeople from sector A, B, and C", or whatever classification of interest you may have).

Your visualization duplicates the "salesman" rows, which worsens the problem relative to adding more salespeople. In addition, if you use color to show quantity, absence of data could be mistaken for 0 (unless you choose different colors for 0 and for missing data and make it very explicit in the legend, but it may require additional cognitive processing for your audience).

Small multiple (or facet grid) with a series of line charts could work, with "salesperson" as main rows and "product" as main columns, and within each cell "dates" as the x axis and "sales" as the y axis. It allows comparison between products and between salespeople, doesn't have the problem of duplicated rows that your original attempt had, and you immediatly spot missing data:

Using lines combined with points allows a more specific identification of the missing data problem. Using only lines without points would generate completely empty graphs for the cells in question, so you wouldn't see for which specific dates there's missing data.

Your visualization duplicates the "salesman" rows, which worsens the problem relative to adding more salespeople. 

In addition, if you use color to show quantity, absence of data could be mistaken for 0 (unless you choose different colors for 0 and for missing data, and make it very explicit in the legend, but it may require additional cognitive processing for your audience). It could be important to make the distinction between the two, e.g. it could be very well that the salespeople did sell products, but the system had a bug that prevented data entry at some point.

Small multiple (or facet grid) with a series of line charts could work, with "salesperson" as main rows and "product" as main columns, and within each cell "dates" as the x axis and "sales" as the y axis. It allows comparison between products and between salespeople, doesn't have the problem of duplicated rows that your original attempt had, and you immediately spot missing data:

Using lines combined with points allows a more specific identification of the missing data problem. Using only lines without points would generate completely empty graphs for the cells in question, so you wouldn't see for which specific dates data is missing.

You could combine the "product" columns into a single column, so each cell would consist of two lines instead of a single one. However, after a couple of tests, I find that the graph feels more cluttered this way, even if I'm not entirely sure why:

You could combine the "product" columns into a single column, so each cell would consist of two lines instead of a single one. However, after a couple of tests, I find that the graph feels more cluttered this way, even if I'm not entirely sure why (maybe because it makes it a bit more difficult to distinguish which 'product' line belongs to which salesperson):