Actually, not really. What you're thinking about is a civilian paramedic. Military field-medics would only carry the basic equipment needed to stabilise the patient in the field (or 'triage'). ODSTs in particular wouldn't be expected to carry a huge amount more gear, much like modern-day Marines or SAS. A field-surgery kit would be enough: floor mat, hemostat set, some bandage scissors, tweezers for removing foreign objects (bullets), a sterile scalpel set, gauze and bandages, some suture sets, some painkillers, some sterile wipes and gloves, perhaps some anti-shock cocktails (hemostil is used nowadays for trauma patients).
So, in terms of an ODST, they'd carry a few field surgery kits (nowadays these kits come in single-use packs to prevent cross-contamination and infection), and from what we know from H3: ODST, some biofoam canisters.
What you have to realise that ODSTs are combatants - much like today's modern military, ODST field-medics would be armed and combat-ready. This means they need to balance their gear between medical and offensive, and given that most medics even today work on a triage system of 'if they can move, they can be treated elsewhere', it's unlikely an ODST would carry bulky medical equipment to do anything more than removing a Spiker round or patching up a plasma burn.
TL;DR: if you want authenticity, take along a few modern field-medic packs. If you need more, just let me know.
Isn't it great working in the medical industry?