National Library of China is the national library of the People’s Republic of China and is one of the largest libraries in the world. It offers some online freely accessible databases of Chinese rare books, such as the 中华古籍资源库; and also the 民国图书数字化资源库 for publications from the Republic era. A list of their digital resources can be found here.
National Central Library, the national library of the Republic of China (Taiwan) provides access to many open-access indexes and databases, see here. Examples include:
抗日战争与近代中日关系文献数据平台 (Data Platform of the Anti-Japanese War and Modern China-Japan Relations) is a platform of digitised books, newspapers, journals and documents relating to China’s modern history. Despite its title, it covers much more than just China-Japan relations.
Modern History database (MHDB) integrates the Modern History Institute of Academia Sinica (中央研究院近代史研究所)’s digital resources and offers one platform to search across all its databases.
National Social Science Database (国家哲学社会科学学术期刊数据库) developed by Chinese Academy of Social Sciences, is the most comprehensive open academic journal database in the fields of social sciences in China.
Library of Congress’ Chinese Rare Book Digital Collection draws from the 5300 titles of Chinese rare books housed at the Asian Division of the Library. It brings together printed books, manuscripts, Buddhist sutras, works with hand-painted pictures, local gazetteers, and ancient maps that encompass a wide array of disciplines and subjects.
Harvard-Yenching Library Digital Collections are listed on their Research Guide for Chinese Studies.
The University of Hong Kong Libraries’ digital initiatives such as their Cantonese Opera Clay Print Libretto collection, Hong Kong Oral History Archives are available at https://www.lib.hku.hk/database/.
The Institute of Oriental Culture, University of Tokyo digitised and offers 汉籍善本全文影像资料库.