What is the legal ruling for a Muslim scholar, who works on innovations to serve Muslims, to ask wealthy Muslims for money to support his research? Is this considered blameworthy begging from people, or is cooperation between the scholar with his effort and the rich person with his money for the triumph of Islam preferable?

The situation mentioned in the question does not fall under the dispraised asking for money, because it is not for the personal benefit of the asker but for the benefit of the Ummah and the realization of a public interest. This differs from asking for a private interest. Al-Hafiz Ibn Hajar, in "Fath al-Bari", indicated that the dispraise of asking for money applies when one asks for oneself, whereas asking for others differs according to circumstances. Moreover, this is more deserving of exception and consideration than bearing the financial burdens (hamālāt) mentioned in the Hadith (definition), which are sums of money a person undertakes to reconcile disputes, and for which asking is permissible.