if a PC wants to make a golem specifically, the various Manuals of Golems have them covered. That's pretty clear in the rules (Dungeon Master's Guide, page 180).
However, there don't appear to be many rules for how a PC can create constructs other than golems, such as animated suits of armor. (Page 129 of the Dungeon Master's Guide talks about how much material, time, and money is needed for magic items of known ratings - common, uncommon, rare, very rare, and legendary. However it is not clear whether animated suits of armor are "common," and possibly each Dungeon Master is supposed to make a campaign-specific ruling.)
(Animate Object lasts only 1 minute. Obviously a construct should be more-or-less permanent, and I'm used to how other editions handle converting temporary effects into permanent ones via a permanency spell or something similar, but permanency doesn't appear to be in 5e.)
So how is someone supposed to create constructs other than golems? The Animated Objects entry in the Monster Manual (p. 19) says:
Animated objects are crafted with potent magic to follow the commands of their creators.
so therefore anything can make one with the correct mind sets to it.
people who are resentful and strong who do things for themselves more as well as less for others. maybe people who only help others in times felt with guilt and or stress, maybe even pain and resentment.
These where all things i personally believe could explain batman's approach in this topic.
Explanation:
The Manual of Golems (DMG, p. 180) requires it's reader to possess at least 2 5th level spell slots, the same as the 5th level slot to cast Animate Objects.
I know no other spell that could turn armor into an animated armor. I asked myself why would someone with access to 5th level spells try to get a CR 1 animated armor when he could get a CR 9 clay golem.
Since this seems like a significantly less powerful use case. If you wanted to house rule it you could allow a Manual of Animated Objects that only requires one 5th level slot and would be a rare instead of very rare wondrous item.Short yes, is up to the DM to decide what kind of potent magic or Manuals are needed to create anything different than the listed Golems
Long The definiton of the Golems states that "...the golem's creator infuses it with a spirit from the Elemental Plane of Earth." so ,in the specific case of the animated armor, I would let the PCs to create minor constructs like that since the moment they can cast Conjure Minor Elementals as far as they have a manual for it.
Personally with the Crawling Claw and the Homunculus (both CR 0) I would let the PCs create them since level 5 or soas far as they can get the hands on the specific manual or get the knowledge from some powerful being (either option would be a source of adventures on its own)
With more problematic constructs like the Helmet Horror or the Shield Guardian, you can compare the CR of the construct with the CR of the creatures conjured with different spells to get an idea of the level that the PCs should have to create them, and/or take a look to the creation rules and requisites of D&D 3.0 or 3.5.
