Give them a scaling, daily use-limited ability to reduce their infusions' casting times
While the artificer is an incredibly strong class, their ability to use infusions rapidly through action points is not, strictly, one of their strengths. Instead, it's more of a quality-of-life thing, allowing them to actually use infusions in cases where they don't have a significant amount of time before encounters to plan and set up their abilities.
Actually giving them action points is an option, but it's one that opens up more doors, or if you don't want other action point-related abilities to be used, requires more exceptions to be written in. I am of the mind that "no action points, but you can do this new thing" will probably feel better to the player than "you can do the thing that all characters get in Eberron, but actually not really, it's just for this one ability." A slight difference, but an important one.
Thus, my suggestion is to give them this ability:
Fast Infusion: Once per day, you can imbue an infusion in 1 round, instead of its normal casting time. At character level 5th and every five character levels thereafter, you can use this ability an additional time per day.
Why this scaling? Action points follow the following progression (ECS p. 45):
\begin{array}{c|c c}
\text{Character Level} & \text{Action Points} \\ \hline
\text{1st} & \text{5} \\
\text{2nd—3rd} & \text{6} \\
\text{4th—5th} & \text{7} \\
\text{6th—7th} & \text{8} \\
\text{8th—9th} & \text{9} \\
\text{10th—11th} & \text{10} \\
\text{12th—13th} & \text{11} \\
\text{14th—15th} & \text{12} \\
\text{16th—17th} & \text{13} \\
\text{18th—19th} & \text{14} \\
\text{20th} & \text{15} \\
\end{array}
The standard suggested adventuring day in D&D 3.5 is composed of four encounters (DMG p. 49), and the way experience math works, it should normally take about 13–14 encounters to level up (or four days of adventuring at a solid pace). This means that you can divide the number of action points per level by 4 to figure out how many you'd use in a day if you were conservative with them. I have placed the break points on the daily uses at a level or two lower than you'd have the action points in question, because this is a more limited usage than action points would have.
This ability would also allow the artificer player to have their infusions truly work on a per-day basis, instead of tracking a "consumable" that relies on the pace of the game (that can be a problem at times, in my experience, especially with campaigns slower leveling than the base game expects by default).
This also would mean that it's entirely possible for the artificer to get more "rapid infusions" per level than they might have with action points. This has some upsides and downsides—while it might, from a technical standpoint, be extra power, it's also a step up in playability. With action points, the player must carefully try to guess how to use them, based on expectations of how fast they'll level up. This doesn't have that, and is probably going to create a less stressful gameplay experience (something the artificer is, given its complexity, in short supply of).
Alternatively, give them action points that can only be used for this class feature
... and bar them from taking other abilities or character options related to action points, such as Wand Surge, Action Surge, or unfettered heroism. This will lower the number of options they can take, and may do so in a way that feels worse to the player than the above. In my experience as a game developer, being giving someone a replacement ability generally tends to be better-received than heavily restricting an ability to make it fit, so this would not be my suggestion.
Or if you want, give everyone action points
Action points were originally printed in Unearthed Arcana, where they were quite a bit more imbalanced than Eberron's revised rules for them. They were used in the setting to help push the tone that the writers intended, but there isn't anything intrinsically tying them to Eberron, and it may be that your game is improved by the addition! It might not be, though, especially if your game has a relatively slow level-up pace, and it's possible that it'll just detract from players' enjoyment by adding one more thing to track.
Even if you do, though, I would still consider disallowing unfettered heroism though. The spell is incredibly strong, especially when used by an artificer with the proper feats or abilities. This is speaking as someone who has, in the past, used, abused, and cheesed the hell out of that spell in actual games. It's fun if the campaign can support it, but it far-too-easily spirals into "I have an infinite amount of this action point-based resource" (dragonmark uses, wand charges, fast infusions, etc).