The chains have no speed
Abilities do only what they say they do
In 5e, spells and abilities do only what they say they do. If the ability gave the chains a speed, it would have said so in the line detailing the chains' statistics.
Each animated chain is an object with AC 20, 20 hit points, resistance to piercing damage, and immunity to psychic and thunder damage.
Not only does the ability not give the chains a speed, but the chains are also explicitly defined as objects. An object does not have a speed unless an effect of ability explicitly gives one to it.
Compare to animate objects
For example, compare the wording of this ability to the spell animate objects:
An animated object is a construct with [list of statistics]. Its speed is 30 feet; if the object lacks legs or other appendages it can use for locomotion, it instead has a flying speed of 30 feet and can hover. If the object is securely attached to a surface or a larger object, such as a chain bolted to a wall, its speed is 0.
Notice even an intelligent construct, a creature, (which the chains are not) is required to specifically have a speed given to them.
The ability is lacking much more information than just the numerical speed
Not only does the ability not list a speed but it also lacks a description of how the chain would move (crawling? flying?) and how they would decide where to go. Remember, these chains do not have any form of intelligence or any kind of sight. So, how could the chains have the capability to even decide where to move?
Also note that the Chain Devil is not granted the ability to command the chains to do anything except attack and grapple (which aren't even commands so much as automatic things that happen under the listed conditions). Even if the Chain Devil tried to command them, the chains have no ears or any capacity to understand speech.
The chains are simply objects that are able to make attacks and grapple when the Chain Devil meets the listed conditions. There is no reason to assume that the intent was to give them a speed and, in fact, the lack of key information needed for this to be possible is a decent indicator that this was not the intent. Thus, the chains have no speed.
Note: Having no speed and having 0 speed are mechanically different. Something with no speed cannot move unless something gives it a speed. Something with 0 speed could move once the 0 speed condition is alleviated or if they are able to get a bonus to speed that would increase it above 0. (thanks @DavidCoffron)
Objects do not trigger Opportunity Attacks
Your assessment is correct. The rules for OA state that (emphasis mine):
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
As you can see the rules are very clear. Unless something is a creature it can't provoke an OA.
Further, the turret created by the artillerist is not a creature but an object. From the UA:
The turret is a magical object that occupies its space and that has crablike legs
Thus, the turret also can't provoke an Opportunity Attack.
It may seem reasonable for a DM to want to change the Artificer turret to a creature (likely a construct is appropriate). However, do keep in mind that this changes a lot more than just Opportunity Attacks. There are a number of spells and abilities that can only affect either creatures or objects or work differently with them.
With the release of Eberron: Rising from the Last War nothing changes
This question was asked about a playtest UA version of the Artificer which has now been officially released in Eberron: Rising from the Last War. The wording of the artillerist turret has changed, it is called an eldritch cannon now. However, it is still an object:
The cannon is a magical object.
Therefore, this version of the artillerist cannon also does not provoke Opportunity Attacks, as explained previously in this answer.
Best Answer
Yes, they can make opportunity attacks.
From the description of the animate objects spell:
If you order your animated objects to "Kill those orcs", they will do everything they can to accomplish the task until it is complete without you needing to expend another bonus action (bonus or otherwise) to do it. This would include making opportunity attacks, because:
On the other hand, if you only command them to "Attack this particular orc, this round", they would perform that task and then wait for further instructions, possibly retaliating if attacked.