Here is a new answer to this question, with two student authors (*):
McMenamin, M. A. S., L. P. Zapata*, and M. C. Hussey*. 2013. A Triassic Giant Amphipod from Nevada, USA. Journal of Crustacean Biology, v. 33, n. 6, pp. 751-759.
As far as I know, the oldest fossil amphipods are from Baltic amber (Eocene) and belong to the genus Palaeogammarus. As for the poor fossil record, I imagine that, given their weakly sclerotized cuticle, they have rather low preservation potential.
Interesting question. I had just been looking into isopod and amphipod phylogeny as I may have found a Lower Carboniferous isopod/amphipod from the Granton Shrimp Bed, Edinburgh. The earliest isopod is from the Carboniferous Essex Fauna at Mazon Creek (Hesslerella shermani Schram 1970). I am unclear at the moment whether the beast I have is isopod or amphipod as I need to do a bit more preparation on it. Other than that, the earliest amphipod appears to be from the Eocene amber. The lack of a fossil record for amphipods may be partly due to misidentification, a lack of articulated fragments, and poor preservation potential in most environments. Some of the Triassic and Jurassic isopods do have a superficial resemblance to amphipods, but I haven't researched this area enough yet.
Here is a new answer to this question, with two student authors (*):
McMenamin, M. A. S., L. P. Zapata*, and M. C. Hussey*. 2013. A Triassic Giant Amphipod from Nevada, USA. Journal of Crustacean Biology, v. 33, n. 6, pp. 751-759.