Final answer:
In the event of a client with portal hypertension vomiting blood, the nurse's first action should be to obtain vital signs to assess the patient's immediate condition.
Step-by-step explanation:
A nurse caring for a client with portal hypertension who is vomiting blood should first obtain vital signs. This initial action is critical to determine the severity of the client's condition and to monitor for shock due to potential blood loss. Checking laboratory values, establishing a peripheral IV line, and obtaining a stat platelet count are important, but assessing the client's immediate physiological stability by checking vital signs takes precedence to inform subsequent urgent interventions.