Final ELF executable files are typically built out of other ELF files, reorganized by the linker. The easiest way, of course, would be to specify ELF as the output format of your assembler.
1) If you really want to do this, you could start with an "empty" ELF file (that you get from compiling or assembling nothing, etc.). Then you could use objcopy --add-section, which allows you to add an arbitrary file as a section in an existing ELF file.
This will create a minimal ELF file:
$ echo "" | gcc -c -o empty.out -xc -
2) Alternatively, you could include your raw binary into another assembly file using something like nasm's incbin, which would then need to be assembled as an ELF.
3) A third option (the best so far) would be to provide your raw binary to the linker, and use a custom linker script to tell it what section to put it in (determined from the input file name). The -b flag before an input file will tell ld what type of file it is. This should let you use your flat binary file.
One of the first obstacles you're going to face is getting the entry point to point to your code. Off the top of my head I'm not sure how to edit that.
There is a Python library, pyelftools that may help you in your quest.