Which of the following items would require sterilization between use?

The correct choice is the rectal polyp snare, as it is a medical instrument that comes into direct contact with mucosal tissues during procedures, which presents a risk of transmitting infections. Sterilization is necessary for instruments used in procedures leading to the breach of body cavities, like the gastrointestinal tract, to ensure all microorganisms are eliminated and to protect patient safety.

In contrast, the blood pressure cuff typically only contacts external skin surfaces and can be adequately cleaned with high-level disinfection rather than complete sterilization. The laryngoscope handle, while important in airway management, usually does not require sterilization between uses when it is not in direct contact with mucous membranes, since it is generally used with a sterile blade that can be disposed of after a single use. The vaginal speculum also only needs high-level disinfection in most settings, as it is used against the vaginal walls and can be cleaned thoroughly between uses rather than necessitating sterilization.

Thus, the rectal polyp snare stands out as requiring sterilization to maintain the highest level of safety in invasive procedures.