No - they're not an insurer, as their website says:
HOW WE WORK WITH YOUR INSURER
We help you provide specialist support to your organisation for your specific medical and security needs via dedicated assistance services that are fully aligned with your insurer.
We have proven working interfaces and processes with an expansive range of expatriate and travel insurers directly providing administrative support for claim submissions when required.
We ensure the relationship between your organisation, International SOS and your insurer is as smooth as possible by using direct billing agreements with insurance carriers.
With us, you can provide real-time medical case information to your insurer, enabling them to verify insurance coverage and achieve their claims management directly on your behalf. Information provided by our assistance specialists’ captures important details that if missed could lead to future complications and unnecessary costs.
Our Approach > How we work with your insurer, International SOS, 4 Dec 2016
What they're saying is that much of the work they do for companies ends up being paid for by that company's insurance.
Though International SOS is not it, your employer certainly will have an insurer, not least because otherwise they (your employer) would be liable for medical and other costs which you might incur as a result of injuries or illness on a business trip. (Under the legal systems of most Western countries, at least).
Duty of Care legislation holds businesses liable for protecting the health and safety of their traveling employees.
Countries in North America (U.S., Canada), Europe (France, Germany, Belgium and Spain) and Australia have comprehensive Duty of Care legislation. For example, failure to uphold Duty of Care obligations in the United Kingdom can result in civil and criminal liabilities not just for companies, but for individuals at those companies as well. In Germany, liabilities for sickness and health-related costs actually extend to family members who may be visiting the expatriate in various host countries.
Understanding and Reducing Business Travel Risks for Employees, Corporate Compliance Insights 11/19/2014 [emphasis mine]
While many travellers make no distinction between the practicalities of their leisure and occupational travel, employers, not the employee, carry the liability for business trips.
Safety and financial issues to consider when travelling for business, Personnel Today, 1 Dec 2012 [article subject is UK jurisdiction]
should I be asking my employer to take out additional travel insurance?
You should simply ask your travel department or HR department for details of your company's travel insurance. Explain that you'd like to check what's covered; if you feel you need an excuse, say that you want to reassure your family that you're covered against accident or illness overseas.
If your employer didn't have insurance then you still wouldn't be liable (because your employer would be, rather than you personally) but if the employer is large enough to have a contract with International SOS then it's pretty much inconceivable that they don't already have travel insurance - such an omission would be an act of major incompetence, and would be a massive risk that International SOS (as well as the company's auditors, lawyers, and even travel agents) would be alerting them to.