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I am creating an empty file...

dd if=/dev/zero of=$SDCARD bs=1 count=0 seek=$(expr 1024 * $SDCARD_SIZE)

...then turning it into an drive image...

parted -s $SDCARD mklabel msdos

...and creating partitions on it

parted -s $SDCARD unit KiB mkpart primary fat32 $IMAGE_ROOTFS_ALIGNMENT $(expr $IMAGE_ROOTFS_ALIGNMENT + $BOOT_SPACE_ALIGNED)
parted -s $SDCARD unit KiB mkpart primary $(expr $IMAGE_ROOTFS_ALIGNMENT + $BOOT_SPACE_ALIGNED) $(expr $IMAGE_ROOTFS_ALIGNMENT + $BOOT_SPACE_ALIGNED + $ROOTFS_SIZE)

How do I use mkfs.ext and mkfs.vfat without mounting this image?

You want to format a partition in a disk-image file, rather than the entire image file. In that case, you need to use losetup to tell linux to use the image file as a loopback device.

NOTE: losetup requires root privileges, so must be run as root or with sudo. The /dev/loop* devices it uses/creates also require root privs to access and use.

e.g (as root)

# losetup /dev/loop0 ./sdcard.img

# fdisk -l /dev/loop0
Disk /dev/loop0: 1 MiB, 1048576 bytes, 2048 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: dos
Disk identifier: 0x54c246ab

Device Boot Start End Sectors Size Id Type
/dev/loop0p1 1 1023 1023 511.5K c W95 FAT32 (LBA)
/dev/loop0p2 1024 2047 1024 512K 83 Linux

# file -s /dev/loop0p1
/dev/loop0p1: data

# mkfs.vfat /dev/loop0p1 
mkfs.fat 3.0.28 (2015-05-16)
Loop device does not match a floppy size, using default hd params

# file -s /dev/loop0p1
/dev/loop0p1: DOS/MBR boot sector, code offset 0x3c+2, OEM-ID "mkfs.fat", sectors/cluster 4, root entries 512, sectors 1023 (volumes <=32 MB) , Media descriptor 0xf8, sectors/FAT 1, sectors/track 32, heads 64, serial number 0xfa9e3726, unlabeled, FAT (12 bit)

and, finally, detach the image from the loopback device:

# losetup -d /dev/loop0

See man losetup for more details.

To create an image with multiple partitions, a solution that doesn't require any fancy tools or root access is to first create the filesystems, then concatenate them.

truncate -s $IMAGE_ROOTFS_ALIGNMENT disk
truncate -s $BOOT_SPACE_ALIGNED part1
mkfs.fat part1
cat part1 >>disk
truncate -s $ROOTFS_SIZE part2
mkfs.ext4 part2
cat part2 >>disk

Then run parted or fdisk to create the partitions.

This approach has the downside that the resulting image won't be sparse.