What is needed to change in this SparkFun solar charger module to reach 2A of current?
I will answer but I want to point out that your panel can only supply 7.15W according to the information in your question. In order to charge a single LiPo cell at 2A, the maximum output power would be 2A * 4.2V = 8.4W. It is physically impossible for output power to be higher than input power. But it is still a good idea to increase the charge rate to 2A. This will allow the charger to extract as much power as it can from the panel.

Here is a link to the datasheet for the part:
http://cds.linear.com/docs/en/datasheet/3652fe.pdf

Here is what you should do:

1) Change R1 to 50 mOhms. This sets the charge current.

2) Change D1 to MBRS340 (available from several vendors). The diode in the Sunny Buddy may get over-loaded if you go to 2A charge.

3) Change L1 to 10uH capable of 2.5 to 3A. E.g., Bourns p/n SDE0805A-100M. It is tricky to calculate ripple current for a solar panel, because Vin max does not occur during Iout max. But 10 uH should give you low ripple current.

I did not study the whole design. I am assuming other things are set up in a reasonable fashion (such as the charge timeout and float voltage). I definitely recommend you review the datasheet, and, if Linear has a reference design for this part, review the reference design schematic and any notes or datasheet available for it.

Good luck!