What is the style called where we draw characters without lifting the brush? Consider this artistic rendering of the 福 (fú, happiness) character:

Four characteristics stand out:

The stroke thickness is constant (like in seal script, 篆书).
The brush is never lifted (pushing to the extreme one tendency of the cursive style, 草书).
The brush never passes twice over the same spot (like in neon signage, as remarked below by Daniel Cheung).
There are no angles.

Has this style a name? Are you aware of an online font which would satisfy at least the second constraint?
连书 (lianshu) comes to mind, but this is actually a completely different animal.
I would just call it 一笔画 (one stroke).
Because there are some famous arts are called in this way. Like:

一笔虎: write the character 虎 in one stroke.
一笔龙: write the character 龙 in one stroke.

And the Eulerian path is also called 一笔画问题 (the problem of how to draw a graph with one stroke)