To solve this problem, we can define it as a linear programming optimization problem. The goal is to maximize the number of gifts sent out within a budget constraint of $1000. We have two types of services: postal service pickups and vans. Each postal service pickup can take 100 gifts at a cost of $50 per pickup, and each van can take 80 gifts at a cost of $40 per van. Additionally, the number of vans cannot exceed the number of postal service pickups.

Let's define our decision variables:
- \(P\) as the number of postal service pickups,
- \(V\) as the number of vans hired.

Our objective function aims to maximize the total number of gifts sent out, which can be represented as \(100P + 80V\), since each pickup sends out 100 gifts and each van sends out 80 gifts.

We have two main constraints:
1. **Budget Constraint**: The total cost must not exceed $1000. Given that each postal service pickup costs $50 and each van costs $40, the constraint can be written as \(50P + 40V \leq 1000\).
2. **Van Limitation Constraint**: The number of vans cannot exceed the number of postal service pickups, which gives us \(V \leq P\).

Additionally, since we cannot have negative numbers of pickups or vans, we also have non-negativity constraints: \(P \geq 0\) and \(V \geq 0\). However, in this context, because \(P\) and \(V\) represent counts of pickups and vans, it's implicit that they should be integers. Therefore, we're dealing with a mixed-integer linear programming (MILP) problem.

Here is how you can express this problem using Gurobi in Python:

```python
from gurobipy import *

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

# Define variables
P = m.addVar(vtype=GRB.INTEGER, name="Postal_Pickups")
V = m.addVar(vtype=GRB.INTEGER, name="Vans")

# Objective function: Maximize the total number of gifts sent out
m.setObjective(100*P + 80*V, GRB.MAXIMIZE)

# Constraints
m.addConstr(50*P + 40*V <= 1000, "Budget_Constraint")
m.addConstr(V <= P, "Van_Limitation")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of Postal Pickups: {P.x}")
    print(f"Number of Vans: {V.x}")
    print(f"Total Gifts Sent Out: {100*P.x + 80*V.x}")
else:
    print("No optimal solution found.")
```

```python
```