I took a somewhat different strategy, which was partially based on feedback from supervisors and my grad school mentor.
The first half was largely the same from site to site and consistent with AcronymAllergy's approach to make a broad summary of my experiences and how those created a foundation for different intervention and assessment skills, abilities, knowledge, etc.
The second half was tailored to each site, but I applied to all AMCs and VAMCs so parts of the second half didn't get changed much (e.g., mentioning goals of working with veterans and a VA career for VA sites, but not for AMCs). The rest of the second half was focused on mentioning specific rotations or experiences offered by that individual site, what I was hoping to gain from it, and how it would mesh with my career goals.
E.g., one of my internship goals was to get experience evaluating and treating trauma and PTSD and specifically with EBTs for them (e.g., PE, CPT, NET). So, for sites that had trauma rotations I would specifically mention this in my cover letter and even bold the rotation name.
I would also, at times, relate the rotations or experiences offered by the site back to my previous experiences. E.g., I didn't have any CBT-I experience from practicum, but had many patients with insomnia and other sleep problems, so I mentioned this in the latter half of the cover letter for sites with CBT-I rotations and how that would help me be more effective across settings for my career since sleep problems are so ubiquitous, even though my goal is not to be a sleep psychologist.
This all helped establish the fit I had with each site, which seemed to pay off since I received 12 interview invites out of 15 applications (the other three weren't a great fit, mostly because they were more research focused and my goals weren't and aren't to be a researcher).