There are two aspects to consider. They are can a VGA->HDMI adapter support 400 pixels wide, and can it support 240 lines.

VGA has no pixel clock. A line is a set of continuous waveforms for R, G, and B. This means that there is no problem with a mis-matched horizontal resolution, except that the edges may be ugly.

If a VGA->HDMI convert supports 240 lines is a bigger question, and the answer is less certain. Some might not support it. Vintage VGA cards could be convinced to do all sort of weird line rates, much stranger than 240, so a good VGA->HDMI adapter should support it happily.