Today I Learned

It depends on your definition of "can". Are his actions allowed by law? No. Will anyone stop Trump from doing them anyway? Probably not.

I also want to make clear, these aren't "Democrat agencies." There aren't formally "Democrat" and "Republican" agencies in the federal government. National political parties are formally private organizations, and local political parties are affiliated with national parties with various levels of control able to be exerted on the local parties by the national parties depending on the specific organizations involved and their relationships. It's all complicated, but the salient point is it's all non-governmental. The agencies Trump is cutting funding from are governmental agencies that generally have greater approval/support from segments of the voting populace that generally lean more Democrat in their voting behavior. There are Democrats that don't support these agencies, and there are Republicans that do. There are also likely people in both parties that support the general cause of the agencies but would prefer they would be run differently or have different policies or regulations. Again, in reality it's complicated and nuanced.

Calling them "Democrat agencies" is Trump applying tribalistic language in his usual divisive way to drum up support from his base. The voting populations that broadly support these agencies generally lean Democrat, but that's not catchy and won't get people angry and vocally in support of Trump. So he calls them "Democrat agencies" to paint a picture that, despite the Republicans having control of literally all branches of the federal government, Democrats directly control these federal agencies (which is not true), and that therefore they are acting against the will of the public, who he represents by definition (which is also not true), and therefore they should be shutdown. It's right out of the fascist playbook, and when the media even just quotes his language, they enable him to define the language of the discussion of his actions, and thus they further help Trump shape the narrative of the shutdown.

Nothing in the shutdown gives him the power to do these things. He was in fact doing all of these things before the shutdown, and he had no legal authority to do any of it then either. He's able to do it because his regime is authoritarian and does whatever they want, and organizations that stand to benefit from this authoritarian regime have spent the last 50+ years systematically subverting the checks and balances that were built into the federal government to prevent this kind of authoritarianism. Complicit politicians in the legislative branch prevent impeachment and removal from office of anyone in the regime that breaks the law, and complicit Supreme Court judges prevent the judicial branch from delivering injunctions or other judicial relief or safeguards from these actions. There are coordinated (even if it's just stochastic coordination) bad faith actors at all levels of power in all branches and offices of the US government. It didn't happen over night, it in fact took decades, but no one stopped it, so here we are.

From the legal definition of "can", Trump in fact cannot do most of what he's doing. But in America laws don't matter anymore, so in practical terms he can do literally anything now.