Should the Nikon D5500 be able to auto-focus the Sigma DG 150-500mm 1:5-6.3 APO HSM lens? Auto-focus works with this lens on my Nikon D40X. On the D5500 it toggles between MF and AF. Video of what I am seeing the monitor's info section. https://youtu.be/IIIFbzEwB90
It appears you are shooting indoors. At f/6.3 there's probably not enough edge light making it through the lens to the camera's AF sensor for the camera focus the lens. If you were in brighter conditions or aiming at very high contrast targets it might work better. The PDAF sensor in any SLR with AF compares edge light from opposite sides of the lens to measure focus.
One of the conundrums facing camera designers is whether to optimize a PDAF system to work with narrower aperture lenses (but with less accuracy due to the narrower baseline of the focus sensor's elements for their positions on opposite sides of the lens) or to optimize it with a wider baseline that can take advantage of wider aperture lenses that allow it to be more accurate and work in dimmer light when used with a wide aperture lens. These wider pairs of lines on the AF sensor won't work at all with lenses too narrow, though.
Some cameras hedge their bets a little: some focus sensor lines are optimized for narrower apertures and others are optimized for wider apertures. When used with a narrower aperture lens, the points sensitive at f/5.6 or f/8 will provide some functionality. When used with wider aperture lenses the points sensitive only at or below f/4 or even f/2.8 will provide more accurate autofocus.
Your D40X is probably designed for all AF points to be usable with lenses up to maximum apertures as small as f/8. Most Nikon AF systems at that time were. Your newer D5500 is probably designed to be usable with lenses having a maximum aperture of f/5.6 or wider.
Switching to Live View might allow the image sensor's CDAF to focus, but there's no guarantee the lens' narrow maximum aperture of f/6.3 won't give the CDAF of the image sensor problems as well.
You need to have the firmware updated at Sigma! extract from an email sent to me :- Thank you for your email. Yes we will be able to update the firmware on this lens for you at a cost of £41.99. If you would like to send the lens to us I have posted a link to our returns form online below.