Yes it supports up to 2 gig, it was added in an update though (at the time of release 1 gig carts were pricey and 2 gig carts were insanely expensive) so make sure you have another way.
"onboard storage" is limited to 256Mbit of NAND for the miniSD version (the original EZ4*) and original EZ4 lite and for the deluxe has 384. The EZ4 lite compact does not do GBA very well (it was built as a cut down cart for DS use). This is intended mainly for roms over 128Mbit in size (a select few including some hacked roms) but can be used for any.
I would try for a miniSD as those adapters can work but have caused several headaches these last few months. Amazon seem to do them for about $10 USD and provided you do not get an absolute piece of junk it does not matter how good your miniSD is for GBA usage as it works from PSRAM or NOR memory which is on the EZ4.
*there are 4 EZ4 models
original, gba sized, miniSD, 256Mbit NOR, 128Mbit PSRAM
lite, DS lite sized (ish), microSD, 256Mbit NOR, 128Mbit PSRAM
lite compact, DS lite sized, microSD, can only run roms up to 64Mbit
lite deluxe, DS lite sized, microSD, 384Mbit NOR, 256Mbit PSRAM.
DS lite sized just refers to the shell- I have been an EZ4 lite in an original DS and GBA SP for years now. If you can grind down (I added a little ribbon to remove it) or refit the shell (bad GBA games go for a dollar or something these days if you need a shell).