To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables and translating the objective function and constraints into algebraic terms.

Let's define:
- $x_1$ as the number of SUV cars produced per day.
- $x_2$ as the number of minivans produced per day.

The objective function, which aims to maximize profit, can be represented as:
\[ 7500x_1 + 4000x_2 \]

The constraints are:
1. The SUV car factory can make at most 5 SUV cars per day: $x_1 \leq 5$
2. The minivan car factory can make at most 3 minivans per day: $x_2 \leq 3$
3. The third-party processor can handle at most 5 vehicles of either type per day: $x_1 + x_2 \leq 5$

Additionally, we have non-negativity constraints since the number of cars cannot be negative:
- $x_1 \geq 0$
- $x_2 \geq 0$

Now, let's represent this problem in the requested format:

```json
{
  'sym_variables': [('x1', 'number of SUV cars produced per day'), ('x2', 'number of minivans produced per day')],
  'objective_function': '7500*x1 + 4000*x2',
  'constraints': ['x1 <= 5', 'x2 <= 3', 'x1 + x2 <= 5', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we will use the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="SUV_Cars")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Minivans")

# Set the objective function
m.setObjective(7500*x1 + 4000*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 <= 5, "SUV_Factory_Capacity")
m.addConstr(x2 <= 3, "Minivan_Factory_Capacity")
m.addConstr(x1 + x2 <= 5, "Third_Party_Processing_Capacity")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"SUV Cars: {x1.x}")
    print(f"Minivans: {x2.x}")
    print(f"Maximum Profit: {m.objVal}")
else:
    print("No optimal solution found")
```