Usually in D&D, one should not be allowed to cut off the limbs of a creature that is still alive (unless your table uses the Lingering Injuries table). However, the picture that depicts the petrified condition has an arm severed off of a petrified creature that is also still petrified. My question is this: if your table does not run with the Lingering Injuries table (DMG pg. 272), can a creature cut off a petrified creature's limbs while said petrified creature is still alive?

It comes down to DM and improvised actions.

You can break off a petrified limp the same as you can cut off a normal one. Neither are specifically covered/allowed by the rules, but there is no logical reason you can't.

It all comes down to if the DM allows dismemberment or not, wether the creature is petrified or not is then future consideration.

I think it also depends on your definition of alive. Considering that the creature becomes solid stone, ceases to age, and any poison's within the creatures system become suspended I would not consider a petrified creature to be alive. For all intense and purposes the creature has become an object that could be magically turned back into a living creature. As an Object you could do whatever you want with it. 

Admittedly, no where in the rules does is specifically say you become an object and it does say 'creature' in the description for the condition but to me anything that has been petrified is functionally an object and should be treated as such, similar to a corpse.

I'd say yes, based on the flesh to stone spell, which has this line:

If the creature is physically broken while petrified, it suffers from similar deformities if it reverts to its original state.

To me that says that RAW, you can certainly do it when under the effect of that particular spell. Personally, I'd use that reasoning to allow it generally, but I realize that's probably getting into house rule territory.

The real pickle (for the DM ruling) would be, if an arm comes off, and the creature reverts to skin, would there be an open wound, or would it be closed?

This entirely comes down to what actions the DM permits in their game. The Lingering Injuries table is intended to apply during combat where a sword swing or other attack might have significant consequences or cause an injury that requires more than a healing spell to cure.

However, the table has no relevance as to whether a DM allows creatures to be dismembered in their game or not. That is a separate question completely aside from the option to have Lingering Injuries caused by combat. Does a DM allow players to dismember opponents or animals or any other creature? It is certainly physically possible but depending on the group might be thematically or otherwise inappropriate (perhaps playing with younger players or those who would be negatively affected by horrific imagery) - so a DM might choose to not allow characters to take potentially heinous, controversial or triggering actions in their game.

Other than that, if the DM allows dismemberment (and this is completely separate from the Lingering Injuries option), then knocking the arm off a petrified creature who is currently stone would usually be considered to be removing the limb. Perhaps it would need to be reattached using stone shape or something similar before the petrification is ended in order for the limb to be reattached. However, this is entirely up to the DM - so ask them :)

We covered this in a game.  

The PCs were hired to rescue a team of junior adventurers.  As it happened, they had fallen prey to a basilisk.  They got three statues out, but one had its arm broken off.  They also recovered the arm, and paid an NPC to use… I forget, Mold Stone or some such spell to “reattach” the arm before they paid another NPC to get ointments of stone to flesh made.

The reattachment worked, sort of.  The man who’s arm had been broken off had his arm, but the melding of stone messed all the nerves up so it was basically useless.  They later found him some magic gloves that gave him use of the arm again.

All of this was house rulings, of course.  I thought the loss of use of the arm made sense… meld stone seems more like pottery making than surgery, but the players were super bummed that the guy still had a lingering disability.  If I did it again, I would probably just have had the meld stone work.

(Their original plan was just to have someone hold the broken arm up next to the statue when they used the ointment… which would have been awful and messy.  Thankfully, they thought better at the last moment and hired out for a stone melder…)

It isn't RAW, but I have ruled in the past that if you can recover the limb and most of the pieces, you can reattach the limb using Mending and a Medicine check before turning the creature back to flesh.  Any missing chips were then treated as wounds that could be healed as normal.  If the limb couldn't be recovered or too many pieces were missing, the only thing you could do would be to use Regeneration to grow it back after the fact. 

I have always treated the petrified creature as an object.