To solve this optimization problem, we first need to define the decision variables, the objective function, and the constraints.

Let's denote:
- \(x\) as the number of SUV cars produced per day.
- \(y\) as the number of minivans produced per day.

The objective is to maximize profit. Given that the profit per SUV car is $7500 and per minivan is $4000, the objective function can be written as:
\[ \text{Maximize:} \quad 7500x + 4000y \]

Now, let's define the constraints based on the problem description:
1. The SUV car factory can produce at most 5 SUV cars per day:
\[ x \leq 5 \]
2. The minivan car factory can produce at most 3 minivans per day:
\[ y \leq 3 \]
3. The third-party processor can handle at most 5 vehicles of either type per day:
\[ x + y \leq 5 \]

Additionally, \(x\) and \(y\) must be non-negative since they represent the number of cars.

This problem is a linear programming problem because both the objective function and the constraints are linear. We will use Gurobi, a powerful optimization solver, to find the solution in Python.

Here's how we can translate this into Gurobi code:

```python
from gurobipy import *

# Create a model
m = Model("Vehicle_Production")

# Define decision variables
x = m.addVar(lb=0, vtype=GRB.INTEGER, name="SUV_Cars")
y = m.addVar(lb=0, vtype=GRB.INTEGER, name="Minivans")

# Set the objective function
m.setObjective(7500*x + 4000*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x <= 5, "SUV_Factory_Capacity")
m.addConstr(y <= 3, "Minivan_Factory_Capacity")
m.addConstr(x + y <= 5, "Third_Party_Processing_Capacity")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found: Produce {x.x} SUV cars and {y.x} minivans.")
else:
    print("No optimal solution found. The problem might be infeasible.")

print(f"Objective function value (Max Profit): ${m.objVal}")
```