To determine if it is possible to transform the triangle with vertices $(0,0)$, $(1,0)$, and $(0,1)$ into the triangle with vertices $(0,0)$, $(1,0)$, and $(1,1)$ through a sequence of 90° counterclockwise rotations about lattice points, we analyze the effects of such rotations on the plane.

### Step 1: Understand the Effect of a 90° Counterclockwise Rotation

A 90° counterclockwise rotation around a point $(a, b)$ transforms a point $(x, y)$ to $(a - (y - b), b + (x - a))$. Specifically, the transformation matrix for a 90° counterclockwise rotation is given by:

$$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.$$

### Step 2: Analyze the Initial and Target Triangle

The initial triangle has vertices:
- $A = (0,0)$
- $B = (1,0)$
- $C = (0,1)$

The target triangle has vertices:
- $A' = (0,0)$
- $B' = (1,0)$
- $C' = (1,1)$

### Step 3: Investigate the Orientation

The initial triangle is right-angled at the origin and oriented with a positive area:
$$\text{Area of Initial Triangle} = \frac{1}{2} \times ((1-0)(1-0) - (0-0)(0-1)) = \frac{1}{2}.$$

The target triangle, however, does not change its area but its orientation needs careful consideration:
$$\text{Area of Target Triangle} = \frac{1}{2} \times ((1-0)(1-0) - (0-0)(1-0)) = \frac{1}{2}.$$

### Step 4: Check Transformations

Each 90° rotation essentially changes the orientation of the triangle by modifying the order of the vertices, but it preserves lattice alignments and relative positions about the chosen point. Importantly, these rotations are rigid transformations, which means they preserve distances, angles, and orientations are changed deterministically based on the rotation point.

### Step 5: Feasibility Check

The task is to match the triangle's orientation and shape through lattice-constrained rotations. However:
- Rotating about the vertex $(0,0)$ and accounting for lattice rules keeps the triangle vertices aligned along lattice lines due to the orthonormal transformation nature.
- No sequence of such transformations allows shifting a vertex from being aligned with one axis to the diagonal of the grid without breaking lattice-alignment constraints.

### Conclusion

Based on the properties of lattice-aligned transformations and the requirement to operate within the integer grid defined by lattice points, it is impossible to perform a sequence of 90° rotations that transforms the given triangle into the desired target triangle without violating its lattice-defined geometry.

Hence, the answer is:
$$\boxed{\text{No}}$$