Local Swiss schools will not be ideal for your children based upon their ages. A quick search of EF threads yields many that detail the difficulties surrounding enrolling tweens in local school, primarily due to language acquisition issues that will likely delay their academic progress at this critical juncture, along with the streaming that will have already taken place for students who are the age of your older child and which is on the horizon for students the age of your younger child.
Further, if your stay in Switzerland is for a short length of time and you intend to return to the US (and your children to US schools) in the next few years, you will should enroll them in an international school. As the tuition will be covered, your most difficult step will be finding the time to schedule visits and any existing wait lists.
If you are staying in Switzerland for the long-haul, local school may be a viable option, but you should consider where your children intend to attend university; it will be a more seamless admissions process for your children to stay on an AP or IB track (in an international school) if they wish to attend a US university.
Do get started soon on your research and visits!