The growing volume of daily disclosed software vulnerabilities imposes significant pressure on security analysts, extending the time needed for analysis - an essential step for accurate risk prioritization.Meanwhile, the time between disclosure and exploitation is reducing, becoming shorter than the analysis time and increasing the window of opportunity for attackers.This study explores leveraging Large Language Models (LLMs) for automating vulnerability risk score prediction using the industrial CVSS standard.From our analysis across different data availability scenarios, LLMs can effectively complement supervised baselines in data-scarce settings. In the absence of any annotated data, such as during the transition to new versions of the standard, LLMs are the only viable approach, highlighting their value in improving vulnerability management.We make the source code of AutoCVSS public at https://github.com/nec-research/AutoCVSS.