Developing a product/service as a result of requirements of existing employer

A is currently employed at company X which in turn is a vendor for company Z. A realized that certain aspects of running the business for Z which X is failing at can be productized/offered as a service. A would like to develop this on their own and make a business out of it.

Is this legal? Will X or Z sue A for IP infringement? How can A leverage their relationship with X or Z and turn them and their partners into customers for A?

Steve gives the correct legal answer, but setting that aside, I'd ask you what is your obligation to the guy paying your salary? Obligation has many different aspects, moral, professional even societal expectation. In the West we have duty to render performance. However I have observed other countries where employee loyalty is so scarce it's past the lost&found bucket right into the dead&buried casket. So depending on a whole host of factors a response could range from

1. X are such screwballs I'll be doing Z a service by starting out fixing the problem on my own
2. I'll honorably tell X but if they choose not to take up a constructive (and profitable) suggestion (first refusal) then fair to seek blessing to strike out on own

How you manage the call+transition will reflect the new relationship between A, X, and Z.

Under Anglo-saxon law (UK, Australia, etc), depending on the work-for-hire doctrine, and what is defined under permissible IP assignments, you are allowed to take know-how but not commercial secrets. So one question is whether your insight is a market secret or not. Value can also be captured in different ways ... eg does company have bonus/compensation scheme for employee ideas. This leads to a broader question of why startups form.

failing at can be productized/offered as a service

Philosophical musings ... why are some people entrepreneurs and others employees? Perhaps inclication, perhaps training but one key element is seeing a customer need. As Peter Drucker puts its, the purpose of a firm is to CREATE and satisfy a customer. Customers have needs, often direct (inputs), indirect (reassurance) or even latent (unrecognised). A good technopreneur can imagine new technology (or often deployment of existing). So spotting a productized service is the essence of technopreneurship. The problem is that the market rewards people who execute so turning the insight into a startup takes a combination of skills, suasion, and (mostly) sweat. For example, the inventors of Guitar Hero spent a decade wandering around losing money before hitting the jackpot with Wii. Technology may be a lever but the problem is that everyone else has access to the same tech and are also gunning hard for the same customer with tight pockets. So hard questions for any founder are

1. how the idea be turned into a value proposition?
2. can you capture part of the value?
3. is it sustainable?

As an aside, the business plan is usually ignored by experienced investors but as the saying goes, the plan is nothing but the planning is everything. It shows that you have thought about the 3 key questions and hopefully the assumptions that you are relying on.

The answer depends in part on your jurisdiction and in part on your contract, written or implied, with your employer, and maybe on the contracts between the other parties, which you may not be aware of (exclusivity clauses, etc).